There are relatively few Internet cafés in this region, most in and around Oxford. Internet access is nearly impossible to find in the smaller villages. In those areas, your best bet will be your hotel, although the general rule seems to that only more expensive properties offer access. Even then, broadband is still relatively rare, and Wi-Fi virtually unheard of.
Mices Internet Shop (118 High St., Oxford. 01865/515955). Wired To (138 Magdalen Rd., Oxford. 01865/727770).