Yes, definitely. Many municipalities and local governments issue bonds to fund various projects (schools, hospitals, infrastructure, etc). You can buy these bonds and in that way invest in these things.
In the US these kinds of bonds are tax-free, i.e.: the income they provide is not taxed (by the US government, may be taxed by your State, check local tax laws). There are also dedicated mutual funds that that is all they invest in, if you don't want to deal with picking individual bonds.
As to mom-and-pop stores - that would not be as easy, as mom-and-pop stores are by definition family owned and you can invest in them only if you are personally acquainted with them. Instead, you can invest in small regional/local chains, that while not being mom-and-pop, still small enough to be considered "local", but are publicly traded so that you can easily invest in them. You'll have to look for these.
You can also use social lending platforms, like Lending Club, which I reviewed on my blog, or others, where you can participate in a lending pool to other people.
You can invest in a credit union by opening an account there. Credit unions are owned by their account holders.