What is someone who weaves tapestries called?

The "tapestry kits" are actually needlepoint and not tapestry. From Wikipedia:

Needlepoint is often referred to as "tapestry" in the United Kingdom and sometimes as "canvas work".

One stitches needlepoint.

What would you call somebody who does needlepoint? You call them a needlepointer. (Variously spelled as one word, two words, or hyphenated.)

But suppose you are mistakenly calling needlepoint tapestry? You probably don't want to use the word needlepointer. The word needlework covers a number of textile arts, so you can call somebody who makes needlepoint "tapestries" a needleworker.

A tapissier. From the French. It's probably pretty rare and most wouldn't know what you were talking about, but I wouldn't let that stop me from using a word especially if it's the only option.