Translation:I put in chicken and vegetables.
Yes - that's right や like と is used with nouns. To convey the same sort of meaning as や but with verbs you would add ～り to the た form of verbs eg. 本を 読んだり、手紙を 書いたり、友達と あそんだり の が 好きです - I like doing things like (hints this is an incomplete list ie. the speaker also likes doing other things not mentioned here) reading books, writing letters and playing with friends.
initially i answered "I put chicken and vegetables in it." because, as you noted, english requires an object for our prepositions. However, as long as we don't translate this as a passive, i.e. "chicken and vegetables were put in," we should be fine.
my preference in this context would be to say something like: "I added chicken and vegetables." to avoid a dangling preposition.
A: "What did everyone put in the box?"
B: "I put chicken and vegetables"
C: "I put beef"
D: "I put butter"
This is a perfectly valid and complete English sentence, although it could be made more natural with the word "in."
English sentences only need a subject and a verb. In this sentence, "I" is the subject, and "put" is the verb. "chicken and vegetables" are the objects and are not even necessary. For a more extreme example, I can say "I put.", in response to "What did you do to the chicken?". You most likely will want to ask me "Where did you put it?" or even say "Pardon? You put? Oh, you put the chicken," but you will understand that I put the chicken at least somewhere, and I did not eat it.
や means and but implies that there are other things which you could have included in your list - for example if you said I eat things like cake and chocolate - you would use ya to show that cake and chocolate are not the ONLY foods that you eat. Whereas と is just simply and with no implication that you are leaving out information. か means or.
Put in chicken and vegetables
I am a Japanese, but cannot understand this. When the subject is omitted in such an independent sentence in Duo, I do not understand a meaning. However, it is possible that it is understood even if the subject is omitted if there is an anteroposterior topic.
Thanks for @Analydiate
I think that Japanese often does not use subjects compared to English. If I and the partner of the conversation recognize the languages of the topic, I can omit the subject for the first time.
だから、私は Who put it?
But in English people don't generally refer to chicken as chicken meat. If you were to say I cooked a chicken - no one would think you were cooking a live chicken, they would know you were talking about the meat. Likewise if you said you were chasing a chicken round the yard, everyone would know you didn't mean the meat.
Yes, they both mean 'and' but とindicates that the list is finite ie. no information has been left out while や indicates that there are other things that could be included in your list which you have left out. The difference is for example とりにく to やさい を いれます - I put in/add chicken and vegetables (that's all that I add.
とりにく や やさいを 入れます - I put in chicken and vegetables (as well as other things.....)