Most firms are unlikely to offer visas for internships given the costs for such a short placement. Exceptions tend to be when they have specific recruitment drives to recruit people from certain countries (typically China/Dubai).
It can cost several thousand pounds just to get a visa, so firms don't really see the value in getting them for a 1-4 week vacation scheme unless there is a true talent pipeline needed.
That is true. It is really disappointing to know though since a lot of firms these days recruit exclusively or pre dominantly from Vacation Schemes and it puts us as international candidates at a huge disadvantage. I am personally facing that problem myself. Being based in Dubai, I have found it to be really difficult to secure Vacation Schemes/TCs despite having a first class degree and securing a distinction in the LPC.