I understand that one should not keep homemade salad dressing that contains garlic for a period longer than a week or so. Does this also apply to using shallots in the dressing?
Botulism is a concern when garlic (or shallot...or even fresh herbs) is stored in oil at room temperature. An acidic environment virtually eliminates the risk, meaning that garlic, shallot, and herbs are safe in a vinaigrette (especially one that separates and has those ingredients mainly sitting in the vinegar layer), for example. Refrigeration makes the whole thing even safer. You might also take a look at this question. In general, shallot should be treated the same as garlic when considering the risks of botulism.
In my fairly extensive experience with vinaigrettes in commercial kitchens, garlic tends to turn well before shallots in vinaigrettes.