Of course you can go to a funeral during your period.
There's nothing you can't do during your period, just be sure that you have the right menstrual protection (pads, tampons, cups, etc.) to keep you protected not just during the service but afterwards at the wake too, often at funerals you stay much longer than intended talking with family and friends about the deceased so be prepared for that.
If you're prone to strong emotions during menstruation you may be more prone to crying at a funeral, but funerals are always emotional and if there is anywhere where you can cry openly without being judged or it being blamed on hormones it is at a funeral. So don't worry about mood swings or strong emotions while at the funeral.
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