Planting lavender, peppermint, and marigolds will not remove rats or mice in your home, nor will those flowers keep the rodents from entering your home. Non-native, invasive species of rats (Rattus rattus and Rattus norvegicus) and house mice (Mus musculus) were first introduced to the Pacific Northwest in the 16th and 17th centuries, so management to keep them out of structures has been and will continue to be an ongoing battle. I support your concern about using toxic rodenticides. However, you are going to have to kill the rodents currently using your home. Setting snap traps along walls and then tenting them or setting them inside a shoebox from which you've removed the smaller ends could be a way to trap with minimal risk to your pets. Alternatively, there are metal multi-catch live traps you could use, or you might have to consider the less humane but very effective 5-gal bucket of water with a wire/spinner. You also must identify 1)where and how the animals are entering your home and then block those entrances and 2)whether rodents are reproducing inside your home or its walls/attic/crawlspaces, etc. and 3)safely clean up the waste - wear an N95 mask and gloves, do not sweep, but instead dampen with 409 or other virus- and germ-killing cleaning solution, wipe, and dispose of the waste in your outdoor garbage can. The article I have attached can help you find those entrances. Once found, block them with very tightly-packed steel wool or attach (with screws) unchewable welded-wire hardware cloth. Expanding spray foam and chicken wire will not be effective because rodents can just chew through those.
If your income limits getting professionals to help with those efforts, you should see whether your city or county social services or health departments can provide some cost-sharing or technical support to reduce rodent populations and rodent incursions into homes such as yours.