It's your call either way, and you don't need to directly allow this
As other answers have already pointed out, this is fundamentally your decision to make as DM for this game. As such, I'm not going to tell you which ruling you should make but instead list some items which bear on the decision either way:
- Players do not get to dictate when rolls happen, nor what those rolls are
You decide when to call for a roll, and what that roll should be. By saying that they're constantly making Perception checks, they are in essence telling you when to allow them to roll, and those instances just happen to be when there's something to perceive.
- Making a Perception check is a specific action, not a general condition
From the PHB section on the Perception skill:
In most cases, you need to describe where you are looking in order for the DM to determine your chance of success. For example, a key is hidden beneath a set of folded clothes in the top drawer of a bureau. If you tell the DM that you pace around the room, looking at the walls and furniture for clues, you have no chance of finding the key, regardless of your Wisdom (Perception) check result. You would have to specify that you were opening the drawers or searching the bureau in order to have any chance of success.
Emphasis mine. Players are supposed to use a Perception check like this more as the culmination of a specific action, and that is the major thing which differentiates the skill check from using the passive score. "I'm looking at, listening to, touching, tasting, and smelling everything in an effort to find anything at all" really doesn't fit that very well.
- The game includes passive scores so that they will be used
A weak counterargument, even setting Rule 0 aside, but this suggested approach makes all passive scores irrelevant through a trivial declaration. You don't have to use the passive scores in your game if you'd prefer not to, but if you grant this request there will be little reason for the characters not to be constantly savvy and prepared for anything that ever comes up.
- Making Perception checks may be time consuming
Passive perception is what's used when your mind is on some other task and you happen to notice something amiss. These players are indicating that they want to check everything for everything, which would naturally involve a great deal of careful, methodical examination of everything they come across.
There are a lot of situations where that flat-out isn't going to work, such as trying to escape from a dungeon under a time limit. In other situations it may be possible but not practical, like trying to beat another adventuring party to a treasure hoard. And the classic approach to stopping players from using time as an inexhaustible meta-resource is to have them risk random encounters while they putter around endlessly-- their slow pace could ultimately prove fatal.
The overarching point of this section is that players have made an argument (whatever you think of it) that their approach should grant them extra success against things like hiding enemies and traps. You can allow that or not, but the game itself doesn't need to leave that dominant strategy in place forever-- you can make nonstop Perception checks expensive for them, changing it from a game-dulling easy mode to an interesting decision they have to make in specific situations.
- Always using rolls can produce worse results than the passive checks, complicating the strategy and its narration
Passive scores always are what they are (never mind ASIs)-- they are a minimum of 5 (base 10, plus a -5 modifier from having an Ability score of 1), and having an Ability score of 1 is going to be rare for a PC. The more typical situation is going to be a passive perception greater than 7.
A consequence of that is that a player could potentially roll a result worse than their passive perception. This may not be a hugely likely or consequential situation in your game, but it's worth pointing out that your players won't necessarily get the benefits they're expecting, and could potentially miss some pretty big things in their zeal to notice absolutely everything. As DM, you can make that more likely in key situations.
- Players don't generally get to declare permanent advantages for themselves
This approach of constant examinations has the effect of making the game mechanically easier for the players, particularly if you do not add other constraints to balance it. Enemies that rely on stealth and ambushes will be automatically nerfed, possibly to the extent that you won't bother using them. Traps become less meaningful, and dungeon exploration less tense. And all because the players just announced that their characters were being super careful.
I love it when players come up with clever ideas, and they get to enjoy the benefits of their cleverness. But this argument of constant, focused attention is pretty weak and poorly structured. I doubt you'd allow it if a player said that, because their Fighter is so well trained with a sword he would be really careful during a fight and so should always get Advantage on attack rolls.
The character stats and dice rolls bridge the gap between what a player/character would like to do and what they are actually capable of doing.