Help them carry their walker up the steps onto the bus.

(What is the part of speech of 'onto the bus'? Is it an adjective phrase or an adverbial phrase? Can we use 'on' instead of 'onto'? Thank you for your help.)

That's a prepositional phrase which is modifying the verb.

... and yes, you can use 'on' if you wish.

I should clarify that. You can use 'on' instead of 'onto' but you'll change the meaning of that phrase. Also, it'll become adjectival instead of adverbial.

The phrase "onto the bus" is an adverbial phrase. It modifies the verb "carry" and provides additional information about where the action is happening.

In this case, "onto" indicates movement or direction toward the bus, specifying that the walker is being carried onto the bus.

Regarding your second question, the preposition "on" can have a similar meaning to "onto," but in this specific context, "onto" is more commonly used. "On the bus" implies that the walker is already positioned or located on the bus, whereas "onto the bus" emphasizes the action of moving or placing the walker onto the bus.

Therefore, in the given sentence, "onto" is more appropriate than "on" as it conveys the intended meaning of someone lifting the walker to put it on or inside the bus.