Can you get a urinary tract infection from having sex while on your period?

Probable, not always: Blood is a favorable medium for bacterial growth and also the flow decreases the favorable acidity of the vagina. These 2 factors make it more likely for sex during menses to cause urine infection. Please, remember that frequent sex alone is a risk factor for uti. The so called "honey moon cystitis" is an example for this.

Yes: Women are always at risk for developing utis as result of sex because intercourse promotes bacteria reaching the bladder where they can multiply in urine which is a friendly nourishing environment for them. Women probably have more bacteria between the labia during menstruation which then makes them more prone to utis. Hence need to urinate after sex to empty bladder of bacteria + growth medium.