Are you located in the United States? If so, then the following is written for you.
I don't know of any specific grants that are available just because someone has been convicted of a felony. It's possible that you might find a group in your area that is sympathetic to the needs of people who have been convicted of felonies who are also disabled and who might be able to point you in the right direction as well as be a proactive participant/supporter in helping you to obtain needed monies to start a business. One such group that I can think of might be the Safer Foundation.
Home - Safer Foundation But, I think that they are only located in certain regions of the U.S. I am sure that there are other such types of agencies located in the U.S.
Even if there isn't a Safer Foundation located or operating in the area where you are located, it would be possible for you to contact them to see if there is a similar type agency around where you are.
If I were you, I would just start filling out the applications for financial aid and grants and see what happens.
http://www.sba.gov/ and
SCORE "Counselors to America's Small Business" | SCORE are also a couple of other excellent resources for finding out about monies that might be available to start a small business.
What type of construction is it that you are thinking of getting into? I did get your private message, by the way.