Is a felony conviction a reason for not being accepted at a police academy?

Uh, yes. I am not bothering to research the answer about whether you qualify legally or not because you will not be hired as a police officer with a felony conviction. If you were testify in a criminal case in which you made an arrest as a police officer, the defense would be entitled to impeach your testimony with your conviction - and that is not going to be happening. Only way you can get rid of the conviction is to get a pardon which almost takes a miracle in Texas.