There is no abbreviation for customer number.
In English there are far less abbreviations than in German

Well, there are two standard abbreviations for 'number':

no.
#

But in this context, what you may actually mean is 'identification number,' which is often abbreviated to

ID

If the word 'customer' is mentioned elsewhere nearby, as in a table or database, you could abbreviate it the second time to 'cust. ID' for a very narrow column.

However, I wouldn't recommend abbreviating the word 'customer' in any other context where it's not necessary to save space. In a business letter, for example, it could be confusing, though 'customer ID' would probably be okay.

Vielen Dank.
Werde wohl ID-No. nehmen.

Ist es dann richtig so?

MfG