Not sure if you can book for someone else - but I believe you can (someone else here may be able to confirm)
The value in the UK for using MR points to pay for travel on Amex's travel booking website is not that great however. I believe it works out at around 0.45pence per point.
Undeniably poor value compared to the return you get from transferring points into Airline and Hotel programs - with the tradeoff that it is arguably a much more flexible way of using points to "pay" for travel (because you are just booking normal fares/hotel room buckets there is no issue with black-out dates and availability if your dates are pretty fixed).
If you are unsure about doing it this way - or find you cannot easily book travel for someone else using the Amex travel website there appears to be a new way to do this (albeit slightly differently)
On Amex's website when you sign in there is a new link I’ve spotted recently on the first “Account Summary” page when you log in. It says “Use your points for a Travel Credit” or something like that (under where your MR points balance is). Clicking this link seems to take you into a page that allows you to use your points to pay off any travel-related transactions you have recently put on your card. They have to be travel transactions by the looks of it – if you don’t have anything travel related on your card it shows you a message to tell you you don’t have any eligible charges available, I think.
So in other words, I believe you could completely ignore Amex’s own travel website – and just book flights/ hotels yourself (either directly, or use alternative travel agencies/ websites) to book travel for someone – use your Amex card to pay for it. Then log into Amex’s website when the transaction is posted and use your points to pay for it through this link. Looks as though you still get the same value for MR points though.
But potentially better if you think you can find better air fares/hotel rates on your own, without going through the Amex travel website.
I think the transactions have to be identified as travel (so Hotels, Airfares, travel agencies, car hire, cruises etc – check the T&Cs on the page).
Hope this helps.