Oh my goodness, I thought you were going to say they were 4 or 5. They are definitely old enough, and I'm surprised they haven't started it themselves as I expect their friends will be buying (or at least choosing or involved in the choosing) their own gifts for siblings, parents and friends. 'I've got my mum a...' is a fairly common conversation.
We started at the age of 5ish (so school age as others say) and with a tiny budget. I used to choose a bath cube and be excited for my mum opening it. A small tube of chocolate liqueurs for my dad.
Once we got pocket money it included our brownie or cub subs, church collection ( to teach us to budget through the week, I suppose) and to 'save up for presents'.
By the age of your children I was choosing surprise gifts for my parents and buying from my brother either from suggestions from my parents or what I thought he might like (hot wheels car). Sometimes I paid with my own money, sometimes subsidised.
Most of the primary schools round here have a secret present room at their Christmas fair (not this year obviously, but your kids are past that age anyway). No adults allowed in. There is a selection of little gifts - bath cubes (first time I had seen them since my childhood!), pencils, rubbers, packets of sweets, notebooks etc. Children pay £50p or £1, choose something for their parent, then take it to the wrapping table where they choose paper, ribbon and gift card and are helped to wrap it. Really, really popular and the children are so excited and proud.
In families I know, each parent does something similar with each child - takes them out alone on a guided buying trip for the other parent and sibling.
To kickstart it with your kids now, I would give everyone a budget (up to £5 or £10 per person) and go to a reasonably neutral place eg. Wilko or a garden centre and have half an hour to choose and buy. Goffer a range of food, books and gifty things, a smallish budget means not too much money wasted on unwanted tat! They need to learn sometime how to choose for someone else.