A typical creature, on their turn “can move a distance up to (their) speed and take one action”. By default, they don’t get a bonus action; those are usually granted by class features, spells, etc..
More specifically, regarding Bonus Actions:
Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action. … You can take a bonus action only when a … feature of the game states that you can do something as a bonus action.
The Hound’s description doesn’t specifically preclude their taking bonus actions, so they would be able to were a spell to grant one (note, though, that Expeditious Retreat has a target of “self”, so getting it cast on the Hound might be tricky).
So, what can the Hound do on their turn? They can move, and they can use their action to attack. What does that mean?
If there’s ever any question whether something you’re doing counts as an attack, the rule is simple: if you’re making an attack roll, you’re making an attack.
— Making an Attack
Clearly, then, they can try to bite their target (as per the dire wolf on which they’re based). But, can they grapple or shove? Yes!
Making an Attack calls out those two options:
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple.
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you.
In both cases, the key bit is that they “make a … melee attack”. Since the Hound of Ill Omen can use their action to attack and both grappling and shoving are types of attacks, the Hound can do so.
Expanding on that a bit, there are a few actions one can take on their turn; the ones called out in the SRD are:
- Cast a Spell
- Use an Object
The Hound of Ill Omen’s description includes:
On its turn, (the Hound of Ill Omen) … can use its action only to attack its target.
thus, it is prohibited from, say, taking the Ready action or attempting to Hide. They are similarly precluded from Dashing as their action. But, since grappling and shoving are both attacks, they’re welcome to do either if they so desire (and the other requirements are missed, eg., that the target isn’t too large).
I will note that grappling specifically “(uses) at least one free hand…”; this GM would generally allow a creature with a free appendage (including a wolf’s mouth) to grapple but, RAW, there’s room to say that a Hound can’t grapple because they don’t have a hand.