Should my PhD thesis be submitted under my legal name?
You need to pick a name that you'll use in your professional life. Changing that name has a cost. At this early stage, the cost is low. So, if you want to use your married name professionally, then change to it now. Otherwise, use your maiden name for the duration of your career. (This isn't to say you can't switch later, just that the cost increases over time.)


  would there be any issues with graduating with an old name? 


It may create a slight administrative burden, in that you may have to prove that your degree certificate is yours (given that it won't match your legal name).