Is it permissible for a husband and wife to perform Hajj when only one sacrificial animal (hady) is offered?

The questioner's Hajj is valid, and there are two types of Hady (sacrificial animal): recommended and obligatory.

The recommended Hady: This is a Sunnah for someone who has not committed a transgression, nor abandoned an obligation, nor is performing Qiran or Tamattu' Hajj. A Jadh' (young sheep) from the sheep family is sufficient for it.

The obligatory Hady: This is due to reasons such as a vow (Nathr), Tamattu' Hajj, Qiran Hajj, expiation for hunting, expiation for removing harm, or abandoning an obligatory ritual.

If the Hady becomes obligatory for both spouses, then one ram is not sufficient for them; rather, each of them must offer a sufficient Hady. An institution or individual can be authorized to slaughter the Hady in Makkah.