Does the COVID or flu shot have gluten in it?

Hi, lynn!
As a rule of thumb vaccines do not contain gluten. These excerpts might help to address your concerns.
"Beyond Celiac encourages all members of the celiac disease community to receive a COVID-19 vaccine as soon as they are able. Both currently approved vaccines require two doses to be effective. You should make the decision to receive the vaccine in consultation with your doctor, who knows your personal health history. The Beyond Celiac recommendation is based on CDC guidelines along with publicly available evidence and the expert opinion of our science team."
"In fact, vaccines do not contain gluten and study authors cautioned that this misinformation is a detriment to public health particularly because the number of people who don't have celiac disease but follow a gluten-free diet tripled between 2015 and 2020."
As always though you should consult with your doctor for their professional advice.
Best regards and I hope this helps.