I've had a tablet stolen and I think I know the person who's got it, however, I'm sure my receipt won't be sufficient enough evidence to prove it's mine and I don't have a record of the serial number.
Is there anything I can do other than write it off and claim on my insurance
Did you buy the tablet from new, how did you pay for it. If it was bought new the shop will almost certainly be able to match the receipt to the computer.
Contact the Police, if they get access to the tablet, you may be able to prove it is your by giving information as to what is on the tablet, unless it has been totally formatted there will be evidence of the previous user still on it.
Speak to the person or a close friend of theirs, mentioning no names, tell them of the “loss” of your tablet and tell them that you are going to get the serial number from the shop using your purchase info.....see what happens.
What sort of device is it? If it is an Android and you have a Google account, sign in to the account and click 'find my phone' (it applies to tablets, too). If you're lucky it will tell you where it is - it is very precise with my tablet.