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I will soon run the Underworld Speculation adventure with my group of players. Two of their characters have flying familiars (an owl and a pseudodragon).
Is there something in the rules that states that a familiar (flying or otherwise) can't be used in water?
There isn't any rule change or anything about underwater fighting and exploration. The party is able to breathe underwater thanks to an NPC that casts water breathing.
Being a familiar doesn't change any interactions with being underwater. The things that matter are whether or not they can breathe underwater and whether or not they have a swim speed.
If they don't have either, as most PCs don't, then you treat the familiar the same as you would any PC when going underwater.
You can read more about the rules regarding underwater movement (PHB, Chapter 8) and underwater combat (PHB, Chapter 9).
Since there is a NPC caster using water breathing on the PCs, that spell could also be applied to the familiars and any other friendly creatures the party would like to join them. If not, then they'll just have to hold their breath via the suffocation rules (PHB, Chapter 8).
As per the rules on swimming, if the creature doesn't have a swim speed, then each foot of movement uses another foot of movement. There is no rule that you can't use your fly speed underwater, just that you use your 'speed'. However, as seen in the Q&A "Can you use the Fly spell to move underwater at a speed of 60 feet?", whether or not you can use your fly speed may be up to you as the DM. The rules aren't clear, so make a ruling as the DM (or talk with the DM, if you are a player) about that.
Reasoning:
According to the movement rules in the PHB about swimming:
While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed.
Further, there is a rule concerning swim speeds for monsters in the MM:
A monster that has a swimming speed doesn't need to spend extra Movement to swim.
The MM has no rules concerning the extra movement cost of swimming. Therefore, it is reasonable to assume that the extra movement cost detailed in the PHB also applies to monsters.
The PHB movement rules do not detail flying. And the MM has no rules concerning the conversion of different movement modes such as flying and swimming.
Even if the the calculation of speed is unclear, this does not prohibit them being underwater. For other familiars that do not fly the answer is clear from the rules cited (i.e. half the walking speed).
