Can I get lip fillers if I have a strawberry birthmark on my lip?

Lip filler can be done with a canula, which is less likely to injure blood vessels, this is passed along the lips from the side.
My recommendation would be to get a consultation with an experienced Doctor so they can examine and palpate the lips to assess where the blood vessels are. They can then advise what the best approach would be and the likely results for you.
There would be a risk of bruising which may be significant, as there is always a slight risk of problems even when using a canula.