Is alloca part of the C++ standard?

No. The answer says it all.

Not only is it not part of the C++ standard, it is not part of any standard. It's not part of C nor is it part of POSIX. Furthermore, allow me to quote from the Linux man page for alloca(3):

The alloca() function is machine and compiler dependent. On many systems its implementation is buggy. Its use is discouraged.

(emphasis added)