I understand that the phrase ごめん、待たせてしまって! means "I am sorry (and regret) that I made you wait". This statement does not look like a request. Why is a て form and not past form た used here?
This seems to be simply a colloquial rearranging of the usual "grammatical" order
This happens all the time. Parts of the sentence are left out and put in after the more relevant parts have been said. For illustrative purposes one might take a very simple sentence りんご（を）食べたよ and rearrange it to 食べたよ、りんご.
First, the て form here is the conjunctive form. As in "I made you wait so...". This is sort of an inverted sentence. Like if you said "This is really good...this pizza," instead of "this pizza's really good.
The 待た is not actually past tense, its 待つ in causative form. The つ becomes た and せる is added, so the word is actually 待たせる. Then, in this example, しまう in て form is also added on so you get 待たせしまって .
The causative form of verbs makes the verb mean "to make" or "to let" someone do that verb. You can read more about them at the link below.