No, they can only be elected to nonpartisan positions. Here is the DoD rule, which is an exception to a blanket ban on active duty members running for office:

Any enlisted member on active duty may seek, hold, and exercise the functions of a nonpartisan civil office as a notary public or member of a school board, neighborhood planning commission, or similar local agency, provided that the office is held in a non-military capacity and there is no interference with the performance of military duties.

Here is the full policy on Political Activities by Members of the Armed Forces. It's very long and very specific.