What is the name of your state (only U.S. law)? Pennsylvania
If a non-custodial parent has never worked and gets only SSI, are they required to pay child support? This parent never takes the child and rarely visits with the child.
SSI (supplemental security income) is a benefit for people who are disabled who have not amassed enough social security credits to qualify for Social Security Disability. (SSDI)
Those benefits may not be garnished for any reason. While I have heard of some very rare instances where a judge has ordered that someone pay child support while having only SSI, the vast majority of judges will not do so, because there is literally no way to actually collect support. The amount of income received is well below the poverty level and well below what anyone can support themselves on, let alone anyone else. People on SSI can generally only survive if they live with family and even then they generally cannot pay their fair share of their own support. As far as I know state child support agencies will not even begin to attempt to collect anything if the parent has SSI, because they know its a waste of their efforts.
Even if you were to get a rare judge who would order child support, it would generally be an amount so low that it would take years just to recover the cost of obtaining the child support order, let alone coming out ahead. Your child is 15 therefore the odds of you collecting enough support to recover what you spend in getting an order would be even slimmer.
And...please do not say "but she isn't disabled". Its harder than heck to get on SSI and the odds of someone being successful in doing so if they do not have an actual disability are slim to none.
If you want to relocate with your child then its up to you to follow the procedures in your state to do so and hope that the courts will rule in your favor. Since your son is 15 and has always lived with you, then unless your son WANTS to live with mom your odds are better than the odds for most parents.
However, you are not going to be able to leverage child support against mom to get her to agree to your relocation, so just do it the proper way, through court.