I just started a new job and found out I have to get bonded. I was convicted of a felony. I did obtain a Certificate of relief of disability. I am a registered voter. I am not sure if I can be bonded.

Of course you can be bonded ( well most likely) if you were convicted of bank fraud, perhaps not to work in a bank.
And the cost of the bond may be higher than someone not convicted. So the company may not want to pay the added bond cost.
You just will not know until you turn in the paper work. But if you were honest about the prior record to get the job, and they already know, then they must beleve it should not be an issue