It is your choice either way, and you don't need to allow it directly
As other answers have already pointed out, this is fundamentally your decision to take as DM for this game. As such, I will not tell you what decision to make, but I will list some elements that influence the decision in any way:
- Players cannot dictate when rolls occur, or what those rolls are
You decide when to request a launch and what that launch should be. By saying that they are constantly doing Perception checks, they are essentially telling you when to allow them to roll, and those instances simply occur when there is something to perceive.
- Doing a perception check is a specific actionit is not a general condition
From the PHB section on Perception Skill:
In most cases, you must describe where you are looking for the DM to determine your probability of success. For example, a key is hidden under a set of folded clothes in the top drawer of a table. If you tell the DM to walk around the room, looking for clues on the walls and furniture, you have no chance of finding the key, regardless of the result of the Wisdom (Perception) check. You would have to specify that you were opening the drawers or looking in the office for any chance of success.
The emphasis is mine. Players are supposed to use a Perception control like this more as the culmination of a specific action, and that's the most important thing that differentiates skill control from using passive score. "I'm looking, listening, touching, tasting and smelling everything in an effort to find something" doesn't really fit very well.
- The game includes passive scores to be used
A weak counterargument, even setting Rule 0 aside, but this suggested approach makes all passive scores irrelevant through a trivial statement. You don't have to use passive scores in your game if you prefer not to, but if you accept this request, there will be little reason why characters aren't constantly ready and prepared for anything that comes up.
- Performing checks can be time consuming
Passive perception is what is used when your mind is on another task and you notice that something is wrong. These players are indicating that they want to check everything for everything, which would naturally involve a great deal of careful and methodical examination of everything they find.
There are many situations in which that plan will not work, such as trying to escape from a dungeon within a time limit. In other situations, it may be possible but not practical, such as trying to defeat another adventurous party in treasure. And the classic approach to keeping players from using time as an inexhaustible meta resource is to put them at risk of random encounters while moving incessantly – their slow pace could prove fatal.
The central point of this section is that players have argued (whatever you think) that their approach should give them extra success against things like hiding enemies and cheating. You can allow that or not, but the game itself doesn't need to leave that dominant strategy in place forever – it can make non-stop perception checks expensive for them, changing it from an easy-boring way to an interesting decision they have. to do in specific situations.
- Always using scrolls can produce worse results than passive controls, complicating the strategy and its narration.
Passive scores are always what they are (no matter the ASIs): they are a minimum of 5 (base 10, plus a -5 modifier for having a skill score of 1), and having a skill score of 1 will be rare for a PC The most typical situation will be a passive perception greater than 7.
One consequence of that is that a player could potentially roll a result worst than your passive perception. This may not be a very likely or consequential situation in your game, but it's worth noting that your players won't necessarily get the benefits they expect, and you could miss out on some pretty big things in your enthusiasm for noticing absolutely everything. As a DM, you can make it more likely in key situations.
- Players generally cannot declare permanent benefits on their own
This constant exam approach has the effect of making the game mechanically easier for players, especially if you don't add other restrictions to balance it. Enemies that rely on stealth and ambushes will be automatically nerfed, possibly as long as you don't bother using them. Traps become less meaningful and dungeon exploration less tense. And all because the players have just announced that their characters were being very careful.
I love when gamers come up with clever ideas and enjoy the benefits of their intelligence. But this constant and focused attention argument is quite weak and poorly structured. I doubt you will allow it if a player says that, because his Fighter is so well trained with a sword that he would be very careful during a fight and should always gain advantage on attack rolls.
Character stats and dice rolls close the gap between what a player / character would do I like do and what they are actually capable of doing.