There have been crisis such as the 1876 tied election, but they still could be solved because Congress has the power to certify the electoral college votes and reject them. Has there been any constitutional crisis where there was no constitutional way to solve it?
What counts as a constitutional crisis can be a bit of a no true Scotsman fallacy. There hasn't (and likely won't) be a crisis that just causes the United States or the rule of law to just not exist. There are a lot of undefined areas in hypothetical scenarios, but they are really only undefined because they are only hypothetical. Passing a law or a supreme court ruling would be the likely response to any of these scenarios becoming real.
Two presidents have ignored supreme court decisions, which likely cause a bit of a crisis at the time. The first was Andrew Jackson allowing settlers on Native American land, and the second was Abraham Lincoln suspending habeas corpus. Several other presidents have arguably ignored rules or laws in a malicious compliance sort of way. A president not following an order creates a sort of crisis where the only remedy is impeachment.
Arguably the practice of Judicial review established in Marbury v. Madison itself was formed by stitching arguments from several places together to say the constitution allowed such power. This was a significant power grab by the Supreme Court for themselves and lesser courts. This could in a way be considered a more or less constant constitutional crisis for the last 200 years, or it could just be the thing that has avoided any real crisis.
Prohibition was more of a lesson learned than a full on crisis, but arguably shows the dangers of creating laws as amendments. The need to create an amendment to repeal the earlier amendment creates a bit of a crisis where there is no support for prohibition but it's still law for a brief period.
The civil war is considered a constitutional crisis, by some, but it really wasn't a big deal from a constitutional law perspective. It was an existential crisis for the the United States though. The constitutional crisis part was related to the reconstruction amendments. Granting personhood to slaves created a whole mess of rights that these amendments were created to address. The extremely broad writing of the fourteenth amendment was to essentially make it impossible to deny that former slaves were American, and therefore avoid giving them voting rights or any government aid.