No sooner did we get ...
In modern English the use of the verb to do plus another verb is used as a way of avoiding having to reverse the normal subject-verb order of a sentence.
The alternative 'No sooner got we ...' is archaic, it occurs in Shakespearean English.
So, in Modern English whenever the normal subject-verb-object would be changed by an adverbial clause or by a question the verb to do is used to avoid this and maintain the normal order.
Do you understand?
As opposed to:
Understand you?
No longer do we use this word order
As opposed to:
No longer use we this word order.