"As I recently browsed the baby aisle of a large store, I noticed several brands of baby oil. As they have for years, each baby oil product included mineral oil as their main ingredient. Most of the baby oil brands had only two ingredients. Mineral oil and fragrance. "Fragrance" on an ingredient label indicates that the fragrance is synthetic. One popular mass market brand markets "Lavender Baby Oil." In herbal and aromatherapy circles, pure lavender essential oil is known to be one of the gentlest essential oils for babies and the natural constituents in lavender is calming and soothing to babies. The popular "Lavender Baby Oil." that is sold under a well known name, however, doesn't include lavender essential oil. Although the name sounds gentle and even natural, there is nothing natural about this mass market brand's Lavender Baby Oil. The "lavender" aroma is derived from synthetic fragrance, and the oil used in the product is once again cheap mineral oil." - https://www.beautybythebatch.com/articles/naturalbabyoils.asp
I wouldnt use Baby oil, Mineral oil is sterile until it touches something but the fragrance may cause problems.
I use coconut oil with my partner after waxing to remove the wax remnants but I am careful not to get any inside her vagina as oils do provide a nice environment for bacteria to thrive because they are hydrophobic they resist her natural defenses, bacteria is typically picked up off the skin or around the anus (which is the source of many a UTI or vaginosis). Most oils do go rancid over time as well, virgin coconut oil does, though refined doesnt. Walnut oil doesnt go rancid either.
Personally Id stay away from oils and get some Pjur extra concentrated silicon lube, it is amazing stuff, will give you everything your looking for from an oil, and leaves the skin fealing really soft and lovely, be sure to have a good clean before hand and also be careful of cross contamination from anus to vagina as mentioned.