There is not going to be a busload of World War II veterans visiting the Internal Revenue Service and the congress is not going to pass a special bill to keep those offices open.
I work in contracting for a Defense Department Agency. So far I have not been furloughed during the shut down. The agency is funded with working capital funds which are not subject directly to appropriated funds. We get our funds from providing supplies and services to the warfighters. The military services' appropriated funds replenish a working capital. The fund has some reserve balance which is being keeps the agency running. That will in a week or two become exhausted. Until then we will do our best to do the boring things that keep the Defense Department functioning.
However one of the things that must be done to make award a contract is check to see that company is responsible. Companies are required to certify that they are complying with various laws and regulations and must have a Taxpayer Identification Number (TIN) to check that they are complying with tax laws and are not in debt to the federal government. New business must obtain a TIN and established companies have their TINs changed occasionally. All of this requires the records be maintained and verified. The vendor registration systems interfaces with the IRS systems and problems must be worked out with IRS employees. Those IRS employees are furloughed so there is a small but noticeable and growing number of companies to whom awards can not be made until things get back to normal.
This is no great emergency, but there are little things like this piling up all over the government as this reckless shutdown continues. The longer it goes the more nuts and bolts loosen, and gears rust out. By the time things really break it is going to be very costly and time consuming to fix.
Head Start, NIH, roads, sewers, and Social Security are more important. Things outside of the favored military are surely in worse shape.