My exhusband and I had four children. Two are left at home (13 and 12) these two were adopted from foster care. Ex has not seen them in 11 years and owes ($68,000) child support. Currently incarcerated for failure to pay support. He has requested that he sign over his parental rights to my partner who has raised the kids for five years to end his support obligation. My partner would love this but we are not married. Would he still be allowed to adopt the kids. We live in the Commonwealth of Virginia. Please help.

No, your partner CANNOT adopt your children even you have theirs father's relinquishment of his rights.

§ 63.2-1241. Adoption of child by spouse of birth or adoptive parent.

A. In cases in which the spouse of a birth parent or parent by adoption who is not the birth parent of a child wishes to adopt the child, the birth parent and his spouse may file a petition for adoption in the circuit court of the county or city where the birth parent and his spouse reside or the county or city where the child resides. The petition shall be the joint petition of the birth parent and his spouse but the birth parent shall unite in the petition for the purpose of indicating consent to the prayer thereof only. The petition shall also state whether the petitioners seek to change the name of the child.
§ 63.2-1201.1. Previously married persons who stood in loco parentis during the time of the marriage may adopt in the same manner as married persons.

A. A man and woman previously married to each other who stood in loco parentis to a child during their marriage to each other, and who could have adopted or readopted the child pursuant to this chapter while married to each other, but whose marriage is void, has been annulled or has dissolved, may adopt or readopt the child pursuant to the provisions in this chapter that are applicable to married persons.

B. An individual previously married to a parent of a child by birth or adoption, and who stood in loco parentis to that child during the marriage, and who could have adopted the child pursuant to § 63.2-1241 during the marriage, may, with the consent of the prior spouse who is a parent of the child by birth or adoption, adopt the child, after the marriage has been dissolved, annulled or voided, pursuant to the provisions of this chapter that are applicable to step-parents.

... my partner who has raised the kids for five years...

The partner appears to be in loco parentis (which means "in the position of parent"), as used in the Virginia statute quoted by GV70. However the statute doesn't provide for a situation like yours, in which the prospective adoptive parent (your partner) is not and was not your spouse. I assume Virginia doesn't have another statute which would cover your situation, or else GV70 would have found it.