Will the clips make my roof leak?

Emphatically... "NO!"

Each of your windows has an outer and an inner seal to it. If the window leaks that means your OUTER seal has failed and in turn that your conservatory frame is probably full on water or fills up when it rains. This means the conservatory builder is at fault and needs to be called back to correct it.

Also, bear in mind that if you traditional blinds with screws it will just make your roof leak in more places as you have more screws. So if you hear from a Salesman “Oh! Don’t get clip in blinds as they will make your roof leak” it is rubbish and you need to ask him why screws which he will use will not make it leak.