We do not recommend the use of home remedies. There are numerous reasons, many of which are summed-up nicely in Penn State's web page (linked above) discouraging the use of these homemade mixtures.

Perennial weeds (those that don't die at the end of the year) like most weedy vines require a herbicide that is absorbed so it kills the roots, if you opt to use a chemical to remove the plant. Otherwise, your only other recourse is manual removal, either by digging out the roots or by repeatedly cutting down all above-ground growth. The mixture you describe is only topical and will not be absorbed by the plant and transported in the sap into roots. Anything that chemically "burns" the foliage will make it look like the plant is dying, but it will not have the same result as successful treatment with a systemic herbicide.

Without photos for identification, we're not sure what type of ivy you have, which might impact what approach for removal you take. Virginia Creeper is sometimes called "ivy," and it's native and won't harm trees (at least, not in the trunk strangulation sense). Boston Ivy is non-native but fortunately not invasive (that is, it doesn't seed into natural areas here in Maryland, even though it grows rapidly and can cover wide areas, like almost all vines do), and if it climbs a tree trunk, it shouldn't cause serious damage. English Ivy is the primary type of ivy that should be removed and managed where possible; it is invasive and can spread into woodlands, where it out-competes wildflowers and climbs and eventually kills trees. Of all the "ivy" plants mentioned in this paragraph, it is the only one that will be evergreen (keeps its leaves in winter). Poison Ivy is native and has good wildlife value but, understandably, gardeners usually want to remove it from the yard; it won't harm trees in the sense that it doesn't strangle them or cause trunk damage. The foliage of any vine, native or otherwise, that blocks too much light from the foliage of needled evergreens like spruce, pine, arborvitae, and hemlock, may cause the affected branches to become bare and die back due to insufficient light. It's a natural response to the crowding by the needled evergreen, but branches that die back due to shading will not regrow, so it can be an aesthetic problem and lead to eyesore bare areas in that plant's canopy.

Household vinegar is not strong enough to kill plants; at best, it would "burn" foliage, but the plant will easily replace those damaged leaves. Acetic acid, which is chemically the same material as vinegar, is much stronger in concentration when it's used as an herbicide, and it needs to be in a labeled pesticide container that has usage instructions to be legal to use in that manner. Because it's a familiar ingredient, some gardeners think of it as a safer alternative to commercial herbicides, but that is not the case: it is highly caustic and dangerous to use without the proper protective equipment, especially for skin, nose, and eye shielding. If it evaporates too much (as it's quite volatile, especially if applied in warm weather), the vapors can damage nearby plants. Enough residues reaching the soil might also alter soil acidity (pH), which can stress or damage the roots of desirable plants as well as impact the beneficial microbe communities that roots rely on to stay healthy.

Salt can be toxic to any plant species (as well as those beneficial microbes) not adapted to naturally salty habitats like shorelines. It can "poison" the soil until enough rain or irrigation water leaches out the excess sodium over time. You do not want to stress any desirable plants in the treatment area further by putting salt into the soil. (Even if it were only applied to leaves, it will rinse off and enter the soil once it rains, or as the burnt leaves fall off and decay.)

Dish detergent and other types of household soaps/detergents are not intended for use outdoors or applied to plants. The "soaps" specifically formulated for herbicide use are designed to work on plants while minimizing environmental impacts of the residues. They strip the waxy coating off of the leaves, which desiccates them and essentially causes them to dry out and sunburn. Just like the acetic acid, it's temporary damage that the plant can evade by growing new leaves to replace them.

Glyphosate, the active ingredient used in many (but not all) Roundup brand products (as well as in some non-Roundup brand products), is systemic, and it binds tightly to soil if the residues contact the ground. This means that it is not prone to leaching in stormwater, moving into nearby plant roots, or otherwise causing problems for plants it does not directly contact. Glyphosate also breaks down into low-toxicity byproducts. As a systemic ingredient, it is absorbed into the plant it's sprayed on (which is why, as with any herbicide ingredient, you need to apply it carefully to avoid contact with desirable foliage) and transported in the sap into roots.

Killing the roots is important for the expedient and thorough removal of perennial weeds, because they can easily withstand one or more removals of their foliage. Contact herbicides, in comparison to systemic herbicides, only affect the tissues they are applied to, so they cannot be moved into roots, which allows perennial weeds to regrow. The impact on the plant is not much different from a browsing herbivore like a deer eating those leaves, or a person cutting them off.

Due to that limited impact, repeated applications of contact-type herbicides would be needed to weaken and eventually exhaust a perennial weed, compared to one (possibly two or three for well-established, very resilient weeds) applications of an ingredient like glyphosate. Triclopyr and imazapyr are two other examples of systemic active ingredients, though each has its caveats, like avoiding use near trees and in areas where water could leach away residues that enter the soil.

If English Ivy is the vine in question, you might need to add a surfactant to a systemic herbicide. These are also referred to as adjuvants or spreader-stickers, the latter name typically being what they are labeled as in the pesticide inventory of a garden center. They are additives that help sprays adhere to waxy, water-repellent leaves, which English Ivy has. Some herbicides might not be compatible with them, or might already include a spreader-sticker, which is why it's important to read the label thoroughly to determine if they can/should be used. Spreader-stickers also have a usage label that should be consulted.

If you prefer to avoid any herbicide, then just cut down all above-ground growth of the weedy vine. If English Ivy has adhered to the trunk/bark of the evergreen tree, then leave it be; the cut pieces left there will die on their own and fall off as they dry out (pulling them off can damage the bark and risk secondary problems for the health of the tree). None of the climbing plants above parasitize the tree (they only hold on with suckers or root-like holdfasts), which means any pieces left attached to the tree as the main stems are cut off at ground level will no longer have a water or food supply, so they starve and die off. (They do not take any moisture or nutrients out of the tree itself.)

Repeated removal of all foliage will kill a perennial weed (be that a weedy vine or any other weed), it just can take a lot longer than using a systemic herbicide treatment. You can think of the roots as a battery, storing energy for future growth. As the leaves photosynthesize, they put energy into that battery. Each time a weed is forced to regrow leaves it lost, it uses-up a bit of that stored energy. If you force a weed to keep regrowing by continually removing its leaves as often as they reappear, you will eventually deplete that battery and starve the plant -- it's using-up energy faster than it can be replenished. This method works for both manual removal (cutting off stems/foliage) and chemical removal (like spraying with a contact type of herbicide, such one of the organic products), but since both tactics take similar amounts of effort, the lower-cost and more sustainable approach of manual removal is more practical than trying a chemical.

How long that starvation eradication process takes to be successful is hard to predict -- it might be weeks, months, or even a couple years for really tenacious weeds -- and you'll need to be vigilant for regrowth and remove it promptly, since the longer weed foliage can remain on the plant, the more it's able to recharge that battery again and set back your efforts. If some of the vine is growing over a property line where you can't remove it all, that foliage is going to make your removal efforts less effective unless the neighbor lets you remove or treat the plant on that side at the same time.

Miri