Kind of an odd Franco-Oriental fusion dressing, isn't it? And you're having trouble with the only two ingredients that give it any kind of Oriental character at all. Don't give up or you'll just have kind of an odd French dressing.
There are zillions of vinegars and, indeed, many varieties of rice vinegar and many varieties of rice wine vinegar. Vinegars have been making themselves out of wines and other alcoholic beverages for thousands of years - it is a natural process in which bacteria in the air combine with the alcohol and turn it to acetic acid. But you needn't start with wine or grain alcohol. Nowadays, vinegars are commercially made from a wide variety of fruits and grains. In fact, any liquid containing either sugar or starch can be caused to ferment and become a vinegar.
Rice vinegars are made from fermented rice. Rice wine vinegars are generally made from the lees (dregs) of wine. Both and widely used in Asian cooking. Both, especially those from Japan, tend to be mild and sweet, and add a bit of acidity to dressings and in cooking. The Chinese red and black varieties are also used as condiments, but still are characterized as mild vinegars. There are also Korean varieties, some of which are much stronger.
You can certainly substitute one for the other in your dressing. You might like to track down both types and even different brands of both types and find out what you like best.
Now, where in California are you that you can't find sesame oil and rice wine vinegar? Surely these products are available in a store near you. You can also find rice wine vinegar, rice vinegar, and sesame oil online.