I have a player (who is a Bard/Rogue) who wants to fire both weapons but while I'm not seeing any rules against it I'm consulting for additional opinions.
Is there anything to consider when doing this?
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The Players Handbook errata has clarified as part of the Ammunition property:
Loading a one-handed weapon requires a free hand
This means you would have to put one crossbow away every turn to load the other. This is not sustainable as after the first turn you have to put away one crossbow to load the other.
You may not even be able to do it the first turn (see below).
The Ammunition property says that loading is part of the attack so it is unclear whether you could even get off that first double attack. (Although I doubt most DMs will disallow pre-loading a crossbow).
Drawing the ammunition from a quiver, case, or other container is part of the attack
The consequence of this is that you need a free hand to even attack with the hand crossbow.
The Attack action only allows for one weapon attack unless you have a specific feature that changes this.
The only other way to gain extra attacks is via Two-Weapon Fighting (in the Combat section of the Player's Handbook or Basic Rules) which requires a melee weapon:
When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand.
There is a feat in the Player's Handbook you could offer your player (usually in place of an ability score increase at the relevant levels) that would allow a bonus action attack with the other crossbow. The Crossbow Expert feat says:
When you use the Attack action and attack with a one handed weapon, you can use a bonus action to attack with a hand crossbow you are holding.
This means you can fire the second one when you attack with the first.
The feat doesn't get around the reloading factor on its own, you would need to handwave that aspect for your player's build to work.
The Crossbow Expert feat does allow one to attack with one hand crossbow twice (Thanks @Carcer) since you only need to make an attack with a one-handed weapon. This has even been clarified officially by WoTC in Sage Advice
A hand crossbow is a one-handed weapon, so it can, indeed, be used for both attacks, assuming you have a hand free to load the hand crossbow between the two attacks.
Attacking twice with one hand crossbow is functionally equivalent to attacking once with two (in fact it's possibly stronger since you keep a free hand) so allowing your player to dual wield via the feat would be balanced. (Just make sure you restrict it to this specific ruling as allowing you to reload without a free hand for any ammunition weapon could have other unintended consequences)