In your sentence, walked away disappointed is perfect to express the fact that the people didn't get to buy tickets, and that they were disappointed.
I looked up to walk away disappointedly and couldn't find instances in GNgram (see below). As an adverb, disappointedly would really have to describe the verb, the way they went away (maybe the manner of walking?). So the adverbs that do follow walk away refer to the way a person walks when they walk away, like slowly for example. It is rather peculiar to say that someone walks in a disappointed manner.
However, it is a different story when other verbs are used, like verbs of speech for example. As you can see in the NGram, said disappointedly is more common than said disappointed. Here disappointedly means in a disappointed manner/tone of voice and this use is much more natural.
This is only an intuition I have, but it seems that walk away is being used in a "linking-verb way", like the verb sound for example.
They walked away disappointed. They sounded disappointed.
Here is a very good way to use disappointedly:
Nevertheless, when we are faced with a choice of evils, very often we have to accept reluctantly and disappointedly that kind of provision. (Example from the Hansard archive. Contains Parliamentary information licensed under the Open Parliament Licence v3.0) - Cambridge
The example you found in the Oxford Learners Dictionary, uses disappointedly in the beginning as a sentence adverb. Thought.co says:
In English grammar, a sentence adverb is a word that modifies a whole sentence or clause within a sentence. A sentence adverb is also known as a sentence adverbial or a disjunct.
"Apparently there is nothing that cannot happen today." -Mark Twain
Fortunately, Ned was invited to a surprise party.
As you can see, they can be followed by a comma, or used without a comma.