Extension doesn't keep databases of companies that would do this type of commercial work and doesn't give recommendations  since that would be interfering with business competition. I suggest you check on some of the sites where you can ask about various companies and which allow individuals  who have hired the company give plus or minus  reviews of past work done.

The best time to remove it is now into July while it is actively growing and flowering.

Although it is a beneficial native plant, most people don't want it to grow where they are likely to come in contact with it themselves. Depending on the size and extent of poison ivy infestations it can take a lot of work to remove. P.I. can travel below ground and spread at a distance beyond the initial emergence. If the roots aren't completely removed or killed with herbicide it will re-sprout.  It takes vigilance to keep ahead of it.  So, you may be talking about asking a company to come in and completely dig up and remove the entire area with the P.I., honeysuckle and Virginia Creeper in order to rid you of it.

New or small sprigs can be pulled up out of moist soil and dumped into a trash bag. I purchased special gloves that go up to my armpits because I am always dealing with the stuff, being next to woods where it grows up trees. Birds eat the berries and deposit the seeds in my yard continuously, so it is a never ending battle. 

Generally vines of good size, especially those that are attached to trees, get cut and treated  on the lower severed end with an herbicide with glyphosate or triclopyr as main ingredient.  Companies that you contact may have specific preferences on how  they choose to deal with it.

Here is our information on poison ivy:

https://extension.umd.edu/resource/poison-ivy/