Recurring canker sores. Do I need special blood tests?

Consult your doctor. Sores in the mouth occur due to many reasons, including, nutritional deficiency, viral infection, lichen planus etc. Please consult your doctor for a proper evaluation and to determine if any tests are needed.
None are specific. There are no specific blood tests to differentiate one form of mouth ulcer from another. There's a short list for recurring mouth ulcers - canker sores, oral herpes (fever blisters or cold sores) & behcet's syndrome. (the panel may add a few i missed). Herpangina & hand-foot-mouth syndrome are one-time ailments. Reviewing your history with an internist or allergist/immunologist is a good idea.