No, a sponsor cannot be on welfare.
I don't think there is a specified amount of time that the sponsor needs to be off welfare, but the vo needs to be confident that the sponsor and their sponsored family member won't be accessing it any time soon thereafter since, all of it will have to be paid back if used in 3 years following sponsorship approval.
If one gets off welfare and gets a job a couple weeks later and then applies, it probably won't be sufficient enough to prove they won't need to go on it again soon in the near future.
If sponsor gets off welfare, gets an ok job, and also shows they have support or living accommodations, etc. Being provided by a family member, then that might be ok.
But I don't think there is a specific formula for the sponsor to follow after they are off of welfare.