“It may depend on the severity of the crime.”
Indeed, that is a tough question. It depends on the mission agency and on the country and on the kind of visa.
When a person applies for a residence visa to live overseas for an extended period of time, they are usually required to present a police report along with school records and job skill credentials. If the criminal record is recent, it is unlikely that you would be granted a visa. If it was 15 years ago, and you have been clearly involved in good things since then, you might get the visa. God is very capable of opening even difficult doors when he chooses to!
It also can depend on the severity of the crime. Sex offenders will have a very difficult time getting a visa to live somewhere. The same with “violent” crimes.
Apart from that, mission agencies would treat it on a case-by-case basis.
My bigger question is “What are you doing now?”
How are you plugged into the church you attend? How’s God using you to minister to others now? How does your pastor and those you work with see what you are doing?
I have two friends who are felons. One I would gladly recommend for ministry. The other I would not. The first relates well with people, is involved in his church and is evidently growing. The second one has issues that come up every time we get together. He isn’t seeing things the way God sees them. His values and comments make people nervous to be around him.
Commit yourself to growing and becoming all of what God wants you to be. Ask your friends and church leaders what they think. People may be reticent to tell you what they think about your involvement in ministry and missions, but if you respond positively to their comments, and try to understand why they are saying what they’re saying, you’ll benefit greatly.
In the end, every mission board will ask for confidential references from those church leaders and friends. You’ll be miles ahead by hearing from them first.