It is normal, although not practical and especially not cheap for the client. I think that the reason is, for one, that banks always want to use their own valuation company (even if it is the same), two, banks usually have got a private agreement with a valuator, I don't mean for illegal things, just that they have an agreement by which they send business to the valuator in exchange of the valuator, as much as possible, evaluating "favorably" so that if the bank wants to give the mortgage it can do it (for example, within reasonable limits, putting the value in 150000 instead of 145000 so that the 80% of it is what the client needs to go ahead).
A valuator is a valuator and supposedly always legal, so one should be able to contract one oneself and send the report to any bank, but any bank will tell you that they don't want the report you have, they will only accept the one given by their own valuator. Also, it is as well a question of them believing in their valuator and preferring not to take risks with some other's report.