All perfect praise be to Allah the Lord of the Worlds. May Allah`s peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
According to Islamic Law, the general rule is that Udhiyah (A ritual animal sacrifice of a livestock animal during Eid Al-Adha) is a Sunna for the Muslim who can afford it and the one who can`t afford it isn`t required to offer it. However, if the latter wanted to offer it and borrowed money to pay for it then it is valid and accepted. An-Nawawi (May Allah have mercy on him) said: "Udhiyah is a confirmed Sunna to be observed by the one who can afford it, but it isn`t obligatory according to Sharia. Since, in principle, it isn`t obligatory, vowing to offer it makes it an obligation, just like any other act of obedience, upon the person making the vow." [Al-Majmou` Shareh Al-Mohathab, Vol.8/P.383].
As for buying an Udhiyah via a credit card, this is permissible if that card is covered.
If it isn`t covered then the amount paid via the card is tantamount to a loan; in which case the purchase is allowed with conditions:
1- The issuer of the card doesn`t stipulate receiving any usurious addition over the original borrowed sum.
2- The issuer of the card doesn`t stipulate any fine in case the customer failed to pay the borrowed amount on time.
3- The trader doesn`t increase the price of the commodity because of the commission that the issuer of the card takes, if any.
In conclusion, generally speaking, it is valid to purchase an Udhiyah by installments via a credit card while observing the above conditions. As for the ruling of Sharia on the system of a certain credit card, this requires reviewing the terms and conditions of dealing in such card in details. And Allah The Almighty Knows Best.