When I do foreplay with my wife, I put my finger into her vagina and take it out, and I do this many times. Do we have to perform Ghusl (a ritual bath)?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

Ghusl (a ritual bath) is only required when a man inserts his penis inside his wife’s vagina, whether they both ejaculate or only one of them ejaculates or none of them ejaculates. The Prophet said: "It is an obligation to perform Ghusl if the man’s and woman’s private parts touch each other (i.e. the husband  inserts his penis)." [Al-Bukhari and Muslim], in Muslim’s narration, there is a following addition: “…even if he does not ejaculate."

However, if the husband does not insert his penis into his wife’s vagina, then Ghusl is not obligatory unless one ejaculates. Inserting the finger –as mentioned in the question- does not necessitate Ghusl, neither for the husband nor for the wife, unless one ejaculates while doing so. It is only the one who ejaculates when doing this, who is obliged to perform Ghusl [but not his partner who did not ejaculate]. 

Allaah Knows best.