Whether a beast can make unarmed strikes or not is something not covered by the rules - as unarmed strikes themselves are almost unmentioned outside of a single entry in the combat section, references in the martial arts ability, a clarification/correction in the PHB errata, and I believe one mention in the MM.
I should also add that somewhere in the rules (I believe it is the monster manual, but I could be wrong) it clearly states that monsters can take any action available to any character, as well as those specifically listed in their monster stat blocks.
Monsters aren't only limited to those particular actions listed in their stat block. All monsters are potentially capable of Dash, Disengage, Dodge, Help, Use Object, or make any Attack action they like (including grapple, shove, and possibly attack with an improvised weapon or unarmed strike) as these actions are available to all characters. In fact, of all the actions listed in the combat section of the PHB, only the cast a spell action specifically requires a special ability to perform.
With that being said, it is subject to DM interpretation as to whether any given monster has the physiology and/or intelligence sufficient to perform any of these actions - and as such, expect table variation. Heck, even in cases where a monster lacks intelligence, they might still be capable of performing some of these actions if properly justified.
I for example, have on occasion had zombies make accidental help (grapple/athletics and shove/athletics) actions when a group of zombies try to simultaneously maul a PC. While two or three zombies in the mob will make grapple and/or shove (prone) actions, the remaining zombies attempts to simultaneously grapple and maul the PC essentially "help" the grappling zombie to keep the PC pinned (granting them advantage), while others make slam attacks or unarmed strikes (clawing and biting), usually while prone themselves (as they aren't the brightest of creatures, and I tend to run my zombies similar to how they work in movies). The zombies themselves are not deliberately working in a cohesive unit, but are still unintentionally helping each other due to their mob activity.
By RAW, there are no clear rules allowing or prohibiting a creature armed with natural weapons from making an unarmed strike. If a PC can make an unarmed strike with a free hand (while holding a weapon in the other), it is possible for some creatures to be capable of unarmed strikes as well - bears being the best example. As such, expect table variation.
