They can be. If they wish to engage in political activity, they can simply separate that from their tax-exempt activities like the NRA does.
There is no valid justification to infringe upon the free speech activities of any noncommercial org with taxation. The 1st Amend prohibits infringement of any kind in matters of association and expression. These activities are not commercial enterprise.
Churches are not subject to taxation per the 1st amendment provision about free exercise. You can't take some of my worship and say it was free; it wasn't free, it was curtailed. And that by act of congress, if they ever get around to doing it.
The NRA is a great organization, nonetheless. I applaud your support of them. :>)
They don't have the protection in the 2nd amendment of specifically being mentioned, though, as the church does in the 1st amendment.