At level 20, a wizard gains the Signature Spells feature. This allows them to choose two 3rd-level spells, and from then on they always have those spells prepared and can cast them once each without using a spell slot between rests.

What happens though, if a wizard reaches level 20 and does not have two 3rd-level spells in their spellbook for some reason? Is the ability suspended until they scribe two such spells in their spellbook?

You Gain Nothing

When you level up, you resolve all the aspects that go with leveling up at that time (though DM's often postpone the actual process of leveling up to the end of the adventure day).

If you have no qualifying spells to choose from, you don't meet the requirements for the feature, which by RAW means they would lose the ability to select spells for this feature. Since Signature Spells are selected when you level up, gaining 3rd level spells later won't be able to retroactively meet the requirements.

The Signature Spells feature says:

When you reach 20th level, you gain mastery over two powerful spells and can cast them with little effort. Choose two 3rd-level wizard spells in your spellbook as your signature spells. You always have these spells prepared, they don’t count against the number of spells you have prepared, and you can cast each of them once at 3rd level without expending a spell slot. When you do so, you can’t do so again until you finish a short or long rest.

If you want to cast either spell at a higher level, you must expend a spell slot as normal.

You gain the benefits of leveling up as soon as you level up, unless the DM house-rules otherwise. The feature requires you to choose two 3rd-level spells in your spellbook as your signature spells; if you are unable to do so, you simply don't get the benefit of the feature. This choice can not be made at some later point.

That said, this is a very unlikely situation; the only way it would be possible is if you picked only 1st- or 2nd-level spells when you gained your 5th and 6th wizard levels, and continued picking only non-3rd-level spells as you continued leveling up. (Or, I suppose, if your spellbook was somehow lost/destroyed after your 6th wizard level and you never chose a 3rd-level spell after that.

Obviously, this would be a terrible idea, as you'd get nothing from the capstone feature.

The situation is impossible without the player causing it

For this wizard:

Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook for free.

In the unlikely event that the wizard had no 3rd level spells in their spellbooks, it would have required a deliberate decision not to ever choose two 3rd level spells so as to meet the requirement for Signature Spells.