You have designed a sentence to defeat the punctuation rules of most style guides, including the one I use, the Chicago Manual of Style. You have an explanatory introductory dependent adverbial clause to tell why your stomach hurts in the independent clause, and a compound subject, a three-element (list), telling what did the hurting
Because of my sensitivity to dairy,1 (your homemade milk,2 cheese,3 and butter) hurt my stomach.
CMS recommends that a comma follow the dependent clause (where you have comma 1) and that a comma separate the list items.
Unfortunately, your reader is likely to parse a four-element list with all three commas as list separators, expecting a sentence like this:
Because of my sensitivity to (dairy,1 your homemade
milk,2 cheese,3 and
butter),4 my stomach hurt.
The reader eventually reaches butter, fails to find comma 4 to end the proposed dependent clause, and realizes that "hurt my stomach" won't support the parse of the longer list. This is what linguist Steven Pinker calls "a garden path," a sentence that leads your reader (at least momentarily) astray.
There are three instances in which a semicolon can help:
When two related independent clauses are not joined by a conjunction:
I am sensitive to dairy; your homemade milk, cheese, and butter hurt my stomach.
When two independent clauses are separated by an intervening adverb thusly:
I am not sensitive to dairy; however, your homemade milk, cheese, and butter hurt my stomach.
When the first clause is long or has its own commas:
Because of my sensitivity to dairy, an annoying condition, which was first diagnosed two years ago; your homemade milk, cheese, and butter hurt my stomach.
Alas, none of these conditions obtains for your sentence. You can search for a more amenable style guide, or you can rearrange your sentence. Here's one way that works minimal change to your diction:
Your (homemade milk, cheese, and butter) hurt my stomach because of my
sensitivity to dairy.
The dependent adverbial clause has been moved to the end, and since it's restrictive (defining why your stomach hurt, not just providing incidental information), no comma is required after the preceding independent clause. And the three-element (list), now at the beginning, cannot be confused for a longer list.