How to say
Thank you for being my first customer.
Is the sentence
Vielen Dank, dass du mein/e erste/r Kunde/Kundin bist.
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Yes, in principle this is correct, though I'd suggest the use of the Höflichkeitsform in a customer relation:
Danke, dass Sie mein erster Kunde/meine erste Kundin sind.
Vielen Dank, Sie sind mein erster Kunde.
It may sound somewhat stilted though in a daily business or shop, as of course sombody has to be number one in the morning. And it may be pretty obvious for instance in the bakery at 6 o'clock.
Guten Morgen, Sie sind der/die Erste heute.
would be more businesslike.