I will give you my opinion about your need, the way I see it, at least. It is not a "ready to use" solution, however, only some ideas about the way to do that.
From what I know, there is no way to track insertion of a row in Excel. So you would require a VBA function to be activated on a button, for example. Actually, there is, see Lunatik's answer.
This function would loop over all rows in your main sheet, and create a new sheet when necessary (you would need preferably a unique id for each client, it could be a simple index on the main sheet, depending on the line).
You would create at first your template sheet, with a specific name, and eventually hide it (to not have it in the visible tabs). When I say that the function would create, it would in fact copy this template sheet and give it a unique name (the id I mentioned earlier). You can find ways to copy sheets at this link.
A second operation to do, would be to put data from the row in the main sheet, to the template sheet (if I understood correctly your requirement), which is not really complicated to do in VBA.