I think more facts are necessary to adequately answer this question. Such as:
(1) Has child support been ordered by the court?
(2) Is there an action pending regarding custody/support? If so, which state?
(3) Is the mother on any sort of public assistance?
(4) Is there any sort of custodial/visitation plan?
Generally speaking, the parents can agree to any amount of child support. And that includes no child support payments (Fam. Code § 3580 et seq.). Under most circumstances, the court will approve such an agreement. But you should get that agreement in writing and file it with your local family law court. Be advised, that a parent cannot forever waive the RIGHT to seek child support. Thus, if you file a stipulation stating that the mother has agreed to not receiving support, it's on the record. And then you won't look like a 'deadbeat' if she tries to get support at a later date.
Your circumstances may be different. You should make certain to have frequent and continuous contact with the children too. And establish a solid visitation plan (i.e. have the children visit during extended school holidays, etc.).
The above are just the general basics. If you provide more specifics, you will probably get a more specific answer...
The prior attorney's answer is very good, and his advice should be followed. Too add to that advice, another thing you can do is contact the department of child support services in the county where Mother and the children live, and ask that they establish a support order.
Please note that it's somewhat of an odd request because the situation is odd. In other words, it's an unusual situation where the non-custodial parent (in this case the Father) wants to pay support and the custodial parent (Mother) doesn't want to take it.
On that note, my guess is that Mother probably thinks that if she accepts support payments, that she will have an obligation regarding Father's visitation rights. That is flawed thinking, as paying or not paying support does not create or abolish custody/visitation rights. Both children and each parent have a right to see each other (e.g., custody and visitation rights), regardless of whether and how much support is being paid. Custody/visitation and child support are two separate issues. It is true that the timeshare percentage can alter the "guideline" support amount when it comes to determining such amount. But, the right of contact between child and parent is not altered by paying or not paying support.
In any event, if Department of Child Support services establishes an order for support, then Father will pay support directly to the State Disbursement Unit, not to Mother. Hence, he will have a record of paying support.
I get the feeling there is more than meets the eye in this situation. Probably best if you sit down with a family law lawyer so he/she can get all the facts to better advise you.