is correct, is
also correct? If yes, has the meaning changed at all?
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(Updated to fold in feedback from Chocolate in the comments.)
The や here
appears to be a shortening of is equivalent to the や in the set construction ～や否【いな】や (~ ya ina ya, "as soon as ~"). The single や means roughly the same thing ("when ~, at the time of ~"). The や construction grammatically requires that the preceding verb be in the 連体形【れんたいけい】 (rentaikei, "attributive form" used to modify a noun), which in modern Japanese is identical to the so-called "plain form", "dictionary form", or 終止形【しゅうしけい】 (shūshikei, "terminal or conclusive form" used predicatively, such as at the end of a sentence).
Consequently, your second sample sentence would not be considered correct grammar.
When translating into English, we would indeed use the past tense form of the verb, such as "No sooner had that kid put it in his mouth, when he threw up." or, "That kid stuck it in his mouth and immediately puked.", etc.
Please comment if the above does not address your question.