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For the first time I encountered "tät" when it appeared in this subtitle to a TV episode:
Ich tät ihr ein paar Schlaftabletten geben.
So I searched DWDS and found many other examples.
Duden says that tun can be used "used to describe the subjunctive". But the subjunctive is "täte", not "tät", again according to Duden. So what is "tät"?
This is a very common form in colloquial German (Umgangssprache). The last e in the first person singular form is dropped.
Other examples:
Ich habe => Ich hab sie nicht gesehen
Ich schlafe => Ich schlaf morgens immer bis um 9
Ich esse => Ich ess das nicht mehr
There are several ways to conjugate the Konjunktiv 2:
Preferred way (newspapers etc.): Ich gäbe ihr ein paar Schlaftabletten.
Umgangssprache (usually not in newspapers etc.): Ich würde ihr ein paar Schlaftabletten geben. In the past, they used to say: "Wenn-Sätze mit "würde" haben keine Würde"
Regional variations (your example): The Konjunktiv 2 with "tät(e)" is mainly used in the south, if you want to know more: https://www.atlas-alltagssprache.de/runde-2/f18c/
As infinitezero correctly says, the omission of the "e" is quite common in German. But the use of "täte" or "tät" in the sentence of your question is stylistically inappropriate. The standard formulation is
Ich würde ihr ein paar Schlaftabletten geben.
Nevertheless it happens quite often that people substitute würde by täte. Perhaps it is a regional phenomenon, but my recommendation is not to use it.
