This has been proposed before, and Republicans always shoot it down along with automatic registration for citizens. It's no secret they want it more difficult to vote, not easier.
The law there, according to a former Governor, is actually on contrast to the State constitution which is why a version was vetoed in 2009:
And your politicians went on to use the nefarious "they" to further solidify the law by making it more difficult to change the State Constitution.
Also just a further note, ifs mentioned in the 2nd article that Oklahoma has very low voter turnout, which Republicans always like. Seems odd that suggesting everyone gets free voter ID's would lead to low turnout as opposed to more turnout.