Mdthbs

Solars have the following weapon attack on their stat block:

Slaying Longbow. Ranged Weapon Attack: +13 to hit, range 150/600 ft., one target. Hit: 15 (2d8 + 6) piercing damage plus 27 (6d8) radiant damage. If the target is a creature that has 100 hit points or fewer, it must succeed on a DC 15 Constitution saving throw or die.

If you defeat a solar, can you take its Slaying Longbow and use it?

It's up to the DM.

D&D 5e Monster Manual p.11, under Equipment, says (emphasis mine):

A stat block rarely refers to equipment, other than weapons used by a monster. A creature that customarily wears clothes, such as a humanoid, is assumed to be dressed appropriately.

You can equip monsters with additional gear and trinkets however you like, using the equipment chapter of the Player's Handbook for inspiration, and you decide how much of a monster's equipment is recoverable after the creature is slain and whether any of that equipment is still usable.

Hence it's entirely up to the DM whether you can take and use a solar's bow.

Strictly speaking, a monster attack entry only defines how that weapon works when the monster uses it. The rules don't define whether the slaying effect is a property of the solar or its bow, or what its stats would be if a player acquired it as a magic item, so that's up the DM too.

It would be an exceptionally powerful item (like the 9th level spell power word kill but it allows a save, so approximately as strong as an 8th level spell for free on every attack), so the DM would be within reason to deny the item on balance grounds.

Also note that you don't get the solar's attack and damage bonuses, but rather your own, as is normal for a character wielding a longbow. You don't get the 6d8 radiant damage, which is clearly stated to be from the solar's inherent Angelic Weapons property. You'd also probably have difficulty using what is presumably a longbow sized for a Large creature.

Rewritten for clarity.

First, the Slaying Longbow is listed under the Actions section, and there is no equipment listed for the Solar, so it is up to the DM whether the Solar uses persistent weapons, or forms them at the time of attack, meaning there may not be any weapon to loot.

As for the Slaying Longbow attack, there are 3 parts, attack, damage, and special:

Ranged Weapon Attack: +13 to hit, range 150/600 ft., one target.

As in any other attacks in 5e, the + to hit is a result of the creature itself and any listed bonuses, none of which are present, so you would only use the range and target traits.

Hit: 15 (2d8 + 6) piercing damage plus 27 (6d8) radiant damage.

As in any other attack, this lists the average, die, and ability modifier damage done by this weapon (dependent upon the creature), but also lists bonus radiant damage, which is the result of one of the Solar's traits:

Angelic Weapons. The solar's weapon attacks are magical. When the
solar hits with any weapon, the weapon deals an extra 6d8 radiant
damage (included in the attack).

This indicates that the weapon itself does not do the additional 6d8 radiant damage, so you would not be able to make use of it.

In addition, there is a special section of the attack not present elsewhere:

If the target is a creature that has 100 hit points or fewer, it must
succeed on a DC 15 Constitution saving throw or die.

Which is likely a special attack trait of the Solar, that is not present within the weapon.

Now, assuming the attack is made with a persistent longbow that can be looted, the only info we have for it is a damage of 2d8 piercing, meaning you would not have access to any special properties.

In addition, because the Solar is of size Large, the longbow might be oversized, and if so, likely would possess the Heavy property:

Heavy. Small creatures have disadvantage on attack rolls with heavy
weapons. A heavy weapon's size and bulk make it too large for a Small
creature to use effectively.

Applying all of the above, if the longbow is capable of being looted, you would have a ranged weapon that does 2d8 + Dexterity modifier damage, and which may be oversized, and/or heavy.

Finally, it would of course be up to your DM if you can loot the weapon, what properties it has, and if it is magical or not.