The story of the "Vitter amendment" and the so-called "Congressional exemption" is a confusing, and so I'll try to clear up some things.
The short version:
1) The Patient Protection and Affordable Care Act included a provision that (by many interpretations, including my own) ends employer contributions to the health insurance premiums of staff and members of Congress.
2) The Obama administration issued a ruling that seems to ignore the law and allow an employer contribution.
3) Sen. David Vitter, R-La., has proposed a measure voiding the Obama administration's creative interpretation, and Republicans want to stick this Vitter measure into a budget deal.
4) Talk of "congressional exemption" is misleading.
Now a longer version:
Early in the Obamacare debates, the big liberal goal was a "public option" -- a government-run insurer. Sen. Tom Coburn, in the Committee on Health, Education, Labor and Pensions, proposed an amendment to force Congress onto the government insurer. The public option died, making Coburn's provision moot.
Later, as the Finance Committee crafted its bill, the exchanges became the heart of Obamacare. Sen. Chuck Grassley wanted to knock members of Congress and their staff off of the Federal Employee Health Benefits Plan (a pretty nice employer-based exchange, in effect), requiring them to use the exchanges (unless they got insurance outside of their congressional employment).
When Grassley proposed this rule, he didn't have actual legislative language -- he just had the outline. It was the office of Sen. Majority Leader Harry Reid who put Grassley's amendment into words. Part of this involved copying and pasting text from Coburn's moot amendment.
The legislative language did not address the question of whether Congress could provide an employer subsidy for staff or members. The law putting Congress on the FEHBP provided for an employer subsidy, but that law was made defunct by Obamacare. If the law doesn't explicitly authorize an employer contribution to insurance premiums, Congress can't give employer contributions.
A few weeks back, though the Office of Personnel Management issued a creative interpretation that allowed an employer subsidy.
Republicans attack this a "Congressional exemption from Obamacare," because other people who work for large employers and are on the exchanges don't get an employer subsidy (otherwise they wouldn't be eligible to use the exchanges).
But it's misleading to call this a "Congressional Exemption": Congressional staff and members are the only employees of a large entity who are pushed onto the exchanges, and so a measure to mitigate that targeted punishment can't logically be called an exemption. If DC passed a law just punishing Wal-Mart, but an amendment weakened the punishment, would that be a "Wal-Mart exemption"?
So there's a very good argument that Congress ought to change the law, and accomplish through valid means what Obama's OPM has tried through executive overreach: amend the law to match the intention of Reid and Grassley, that Congress be eligible for employer subsidies.
On the other hand, Obamacare hurts lots of people -- why should Congress be first in line?