We get this too. For a while we would give in a little. Somewhat. But he had to choose.
I think the classic example, was a time when we were on holiday. There are always certain characteristics about toys or other objects that attract difficult child 3. Puzzles, video games, objects which fall or can be shot out of a gun; lava lamps and other paperweight things with bubbles falling or rising - you get the idea. Every day while we were on holidays we got "I want, I want" and every shop we browsed in, we got tantrums on leaving if we hadn't bought him something.
So we finally said to him, "We will buy ONE toy a day. One. Once we have bought the toy for that day, that is it. You have to choose."
Of course there were still problems, difficult child 3 would get us to boy the toy for the day then see another he HAD to have in every other shop. But because we were staying in the same area, we were able to go back to shops the next day. difficult child 3 was learning to choose, which was the first step. After a two week holiday, we had accumulated a lot of stuff. Of, and generally they were family possessions and not difficult child 3's.
Another thing we did was allow a certain amount of money per child, and they had to budget. difficult child 3 was still a bit too young to be able to do this, and his anxiety at possibly missing out on a toy would have him too frantic to think straight. So the first idea above, is what worked best for him at that stage. And it did NOT continue after we got home from holidays!
The same anxiety of "What if it's sold by the time I have saved up my money?" was, interestingly, a problem for easy child. So we instituted "family shop". This meant that we would buy ONE toy per child, to put in our "family shop". Once that toy was bought, it meant it was safe form being sold out form under them. But they had to save up and redeem that toy from us first, before we would put another toy in the shop. The only exception was if there was a going out of business sale, or "We won't be stocking your favourite product any more" sale, as when the local store stopped stocking Sylvanian Families and easy child had been collecting them all. Which reminds me - she now has her own house, and we still have the dolls house with all the toys. Hmm...
The family shop helped settle the "What if they sell it to someone else before I get back here with my saved up money?" but they would still want other stuff later, and have to have me say, "But you already have something waiting for you in the family shop." Having a toy waiting for them did help, but if tye said, "I don't want it any more," then we said, But you begged me to buy it. I spent money which is YOUR money, at YOUR request, and you have to pay me back so I can afford to buy the next toy on your behalf for the family shop." It as a hard but necessary lesson. But the anxiety was always reduced a little by the knowledge that I had something in reserve, even if it no longer seemed so attractive now it was 'safe'.
Try a combination of these, whatever works. But with a difficult child especially one who has anxiety issues, you will probably find that the blanket "We won't buy it" risks not teaching them anything except how to covet, and how to become a hoarder.
Marg