I usually see these words paired with "at subject." Even dictionaries have it this way:
They stood gaping at the pig in the kitchen.
https://dictionary.cambridge.org/dictionary/english/gape
Same with "gawked" and "beamed." (I'm not very sure, though.)
Would it be strange to write this words without an "at subject"?
Example:
He gaped, then told told Lily, "What are you doing?"
Yes, you can gape (or gawk or beam) at something, or just gape/gawk/beam in general.
For gaping or gawking, usually if you don't specify the subject, it's understood from the context, though you can just beam in general without aiming it at anything particular, in which case you're just describing having a huge smile. ("Sarah bounced into the room, beaming.")
Example:
"I'm going into the forest tonight," Harry said.
Ron gaped. "The forest? At night? Are you mad?"
Since it's a conversation between Harry and Ron, it's implied that Ron is gaping at Harry, even though it isn't stated.
The verbs are ways of looking at someone, and they use the same preposition as "look at", just as do "scowled" and "frowned" and "smiled".
It can be correct to say "gaped" or any of the other expression words mentioned without an indirect object (at someone).
Used that way, you aren't specifying who the expression was directed toward. That could mean that it wasn't directed at anyone (you can smile when you are alone), or that it is understood who the expression is directed at, for example, in a conversation between two people.
It's quite normal to use these words on their own. Some common examples
Don't stand there gaping
She beamed all night
I met her, and gawked all through dinner
And then I just gaped
You're correct that "gaped at something" is more common, but it's completely normal to use them alone.
