The U.S. Congress has independent power to declare a finding of "in contempt" for those who do not cooperate with their investigative ability. This leads to a few possible solutions. (It's all over our news, so I know a LITTLE about it.)
They can give the contempt case to the U.S. Department of Justice and have it tried as a federal case. But in this case, that means the DOJ would have to arrest its boss and there are folks who suggest that even if they did, it would take so long that it might overlap into the next election, at which time some of the House members might not still be in office.
They can have the master-of-arms of the appropriate chamber arrest the person and detain him, holding that person in a jail until he chooses to comply with the request for which the contempt proceedings were filed.
There is an option that is so unclear (called "inherent contempt") that I have no idea what it leads to, but it isn't good. It was last used in the early 1800s and wasn't popular then, either.
EDIT: I did some more reading. Inherent contempt is actually related to the fact that Congress itself can hold a trial rather than rely on the DOJ to do it. It was used more recently than I first thought, but did not result in conviction.
Here is the problem: The law regarding the special counsel's report makes it clear that it is NOT to be released because of sensitive content and sources. So when Congress demands the unredacted version, they are themselves breaking their own law. In essence, we call this "entrapment" and there is a remedy for it that involves allowing the entrapped person to disobey one of the entrapping laws. The attorney general has obviously made his choice, but the House committee doesn't seem to like that choice. One thing that this will underscore is that old rule that says the only folks who get rich in government are the politicians and the lawyers who defend the politicians.
EDIT #2: A really huge stumbling block before Congress is that if they refer the contempt case to the DOJ, they (DOJ lawyers) have prosecutorial discretion to not try the case. Also, there is the issue that the power of Congress to issue such a citation depends on determining whether they are investigating someone in a matter over which Congress actually holds power. This relates to the "separation of powers" clause of the U.S. Constitution that keeps the three branches from stepping all over each other toes all of the time in jurisdictional dispute.