Does a sexual disease cause me to miss my period?

Probably not. No sexually transmitted disease (STD) is likely to cause a missed period. Other causes are much more likely. But if you think you are at risk for STD -- new partner, partner with other partners, etc -- then for sure get checked. Testing is easy anc quick. Better safe than sorry!
Unprotected sex. Unprotected sex is great way to catch sexually transmitted diseases as well as to become pregnant. So while STD might not cause you to miss your period, pregnancy certainly could do so. Best way to find out is to take home pregnancy test.