Behavioral economics research suggests paying people might backfire. If you're offering money to get someone to do something, they are likely to conclude that it must be something that they wouldn't otherwise want to do, so you might actually increase resistance (especially among the hard-core anti-vaxxers who would see it as yet another conspiracy). Ironically, the fact that vaccines are currently hard to get may be driving up demand for the same reason-- anything scarce must be worth having. Not rational on either count, but humans aren't.
I think what we do need to be doing is talking up what vaccination will allow you to do again-- as it is becoming increasingly clear that it will be pretty safe for vaccinated people to socialize with other vaccinated people (greatly reduced transmission, not just illness), that is a major selling point. I do think it is a mistake for super cautious public health folks to keep saying you can't change your behavior at all even after you are fully vaccinated. Instead, tell people what they can do that is relatively safe IF they are fully vaccinated.
There's the Israeli kibbutz story of parents not picking up their children on time even after a fine for lateness was imposed. Well, it was a small fine, so one can then rent child care at a low rate. If the fine were a sufficiently high share of income, this wouldn't be observed.
Same applies to vaccines: Give them enough and they will come! And anyway, it doesn't have to be everybody who is sensitive to price.
The additional point about bribing people to take vaccines, and not just to pick up their kids, is that those who don't take vaccines also benefit. By paying people for getting vaccinated we are merely recognizing that they are doing good for others.