One source of such an incentive could be a cap-and-trade emissions trading scheme with strict caps [0], in which air-capturers are granted emissions rights they can sell.
The EU's ETS [1] is currently the biggest carbon market, regulating about 40% of EU emissions.
It does not grant rights to capturers that take CO2 out of the atmosphere - only to those that reduce the emissions at some plant or factory. Apparently this choice is partly because it's not clear how to make sure that the carbon sink is stable - that the CO2 won't get out again. [2]
Lets assume you could get emissions rights for removing CO2. Would this provide the incentive we want?
The current price for emissions rights is 15€ per 1000 tons of CO2 [3], so an operation to sequester all of the EU's emissions (3.4 * 10^9 t, 2014, [4]) would get a revenue of at most 50 million euro / year. Probably much less since the market would get flooded.
Why is this price so low? The ETS' cap could be too low. Also, the ETS does not really cap emissions at all. Only certain industries, like power or aluminium manufacturing need to present certificates [1b], hence the 40%. Also, industries which would be 'at risk' to leave the EU if they were forced to pay, get free certificates [5] - arguably these would pay the most for rights.
Overall the EU does seem to be doing something right though, since the overall emissions are going down [4]. But the ETS does not seem to internalize the carbon externality.
[0] https://en.wikipedia.org/wiki/Emissions_trading
[1] https://ec.europa.eu/clima/policies/ets_en
[1b] https://ec.europa.eu/clima/sites/clima/files/factsheet_ets_e...
[2] https://en.wikipedia.org/wiki/European_Union_Emission_Tradin...
[3] https://www.theice.com/products/197/EUA-Futures/data?marketI...
[4] https://data.worldbank.org/indicator/EN.ATM.CO2E.PC?location...
[5] https://ec.europa.eu/clima/policies/ets/allowances/leakage_e...