They could do it through community land trusts, which is basically where a nonprofit owns the land but leases it to the tenants at very low rates with the purpose of keeping housing affordable in perpetuity. One of the ways community land trusts get land is by having rich people donate it to them and then deduct the value of the land from their taxes. One reason coop housing doesn't get built as much (in the US at least) seems to be that coop housing used to get more funding from local governments, but local governments have favored low income rental housing construction for several decades, because of these getting tax credits that housing cooperatives don't.
Limited Equity Housing Cooperatives
Tax Credit