Your example does not make things clear.
Because links phrases/clauses/sentences, with one part giving the reason for the other part.
Since can be used either of time or of reason/cause.
For example, you can say either:
I took my umbrella with me because rain seemed likely.
I took my umbrella with me since rain seemed likely.
Here since is employed in the sense of because - meaning for this reason.
**Since is also used to specify something that has or has not happened from a specific point (day, month, year, season, period) in time.
I have not seen them since Christmas.
In your example you are implying that having finished your homework, you are now free to accompany someone. But you do not make this clear. To be clear, using either because or since, you might say:
Since (or because) I have now finished my homework and am free, may I accompany you?
You could also reverse the sentence and begin with May I accompany you....