That sequence of words doesn't mean anything! Your problem is that you are dividing the sentence into the wrong chunks. Look at it like this:
In future, I would make sure I'd checked out [whoever was fitting it] with their previous customers.
The noun phrase 'whoever was fitting it' is the direct object of the sentence. 'Whoever' refers to the person who fitted the faulty ventilation system - let's call them the 'fitter' - and 'it' is the system.
The noun phrase 'their previous customers' is the indirect object, and it refers to the fitter's previous customers.
The writer is saying that in future they would contact the fitter's previous customers, and check with them whether they were satisfied with this person's work or not.