Some of the possible solutions I'm about to propose only work for certain methods the Imposter could be using for the deception; for example, they might have replaced the scroll your players are using with one under their control, or they might have stolen the original other scroll. There are too many possibilities to discuss them all, but I'll try to briefly mention what kinds of deceptions each potential solution will be effective against.
If the scroll you have was ever in the Imposter's possession, then the Knowledge Cleric's Visions of the Past feature should reliably help identify them. The relevant portions of the feature are:
Starting at 17th level, you can call up visions of the past that relate to an object you hold or your immediate surroundings. You spend at least 1 minute in meditation and prayer, then receive dreamlike, shadowy glimpses of recent events.
Object Reading. Holding an object as you meditate, you can see visions of the object’s previous owner. After meditating for 1 minute, you learn how the owner acquired and lost the object, as well as the most recent significant event involving the object and that owner.
Your party doesn't have a Cleric, let alone a 17th level Knowledge domain one, but perhaps they could go on a quest to find one and earn their services.
If the Imposter is nearby, such as in the same city, it may be possible to use Locate Object to find the other half of the pair of scrolls. This is a bit situational and requires a bit of DM ruling, but I would probably allow it. Here is the relevant text:
the spell can locate the nearest object of a particular kind, such as a certain kind of apparel, jewelry, furniture, tool, or weapon.
This spell can't locate an object if any thickness of lead, even a thin sheet, blocks a direct path between you and the object.
The DM would have to rule how specific a "particular kind" of object can be, but if you put your own scroll into a lead-lined box then locating another of that "kind" of object should be possible. Since it's a 2nd level spell with no costly components and it's available to your Paladin and Druid, and possibly your Wizard, you could cast it multiple times in a day to cover a larger city. It would be very impractical to search much further than that though.
Scrying is very dependent on the DM ruling that "whoever is writing these messages" is sufficient to target someone, but if it's allowed then Scrying should work on a wide range of possible Imposter methods. There are some complications though. Scrying requires a focus costing 1000gp, which might be quite expensive for your party. It also allows for a saving throw, and since we're already stretching whether the target is even valid they would definitely get the bonus to their save from being not very familiar. Additionally, a successful save means you can't try again for 24 hours. Finally, your party won't gain access to the spell until level 9. All that together means it could take quite a while to actually get a result, though nothing about the spell's description indicates that the target will know about attempts whether successful or not so that might not be a problem.
When players want to do something that doesn't necessary have official rules, it's appropriate to call for a skill or tool check (or technically an ability check that lets you add your proficiency bonus if you have proficiency in a particular skill or tool). Here are a few examples of checks that could be appropriate based on examples from the PHB section on ability checks and the Xanathar's Guide section on tool proficiencies with some emphasis added:
An Intelligence (Arcana) or Intelligence (Forger's Kit) check to determine that someone has messed with the magic of the item directly:
A forgery kit can be used in conjunction with the Arcana skill to determine if a magic item is real or fake.
An Intelligence (Investigation) or Intelligence (Forger's Kit) check to determine an Impostor has forged the scroll itself or is imitating your contact's handwriting:
Investigation. When you examine objects, proficiency with a forgery kit is useful for determining how an object was made and whether it is genuine.
A Wisdom (Insight) check to determine whether the contact has changed writing style subtly that could indicate they were replaced:
Your Wisdom (Insight) check decides whether you can determine the true intentions of a creature, such as when searching out a lie or predicting someone’s next move. Doing so involves gleaning clues from body language, speech habits, and changes in mannerisms.
Your players could also seek help from a Forger NPC if they aren't proficient in relevant tools and skills.
This option is the most dependent on DM rulings, and as a consequence has a lot of potential flexibility and could even be used for more indirect Imposter situations such as mind control or shapeshifters that even Scrying might miss because the writer doesn't look different. Some examples of divine help are the spells Divination and Commune, and the 10th level Cleric feature Divine Intervention. All of these have the potential to gain relevant information from a deity with the right wording, but it's up to the DM exactly what information the deity has and how they choose to share it.
None of your players naturally has access to these even at higher levels, so if you want these to be an option then you'll need to decide how they're accessible as well.