You can make your own mayonnaise without vinegar. I just substitute an equal amount of lemon juice for the vinegar and have never had a problem. Actually I prefer the flavor of the mayonnaise with lemon juice. I thought making my own mayonnaise would be a pain but after you get used to doing it, it is super easy. You really have to have a food processor though.
The downside is that you have to take your own mayonnaise with you everywhere. I purchased a package of small condiment type cups with screw on lids that were designed specifically for this purpose. They are great and even though they were pricey ($7 for 4 at Harris Teeter) I have definitely gotten my money's worth out of them. The advantage of making your own mayonnaise is that you can use good oils like olive or canola and you can be assured of what goes in it. You can also add other flavorings if you like. I'll be honest; I do add sugar. I figure the amount is so small (1 tsp) that it's going to be negligible in the big picture. I am very limited in my eating choices since I have other dietary issues (gastroparesis/SIBO and a diet for Rheumatoid Arthritis ). Mayonnaise is almost essential since a big part of my diet is salad based and traditional dressings are out because they all contain either dairy or high fructose corn syrup and I am both dairy and fructose intolerant.
Oddly enough I have never had mayonnaise spoil on me. If I think it is going to be a hot day, I may take a lunch container with a cold pack, but mainly I have taken it in my purse and have not had an issue. I hope you'll consider making your own. It really is easy.