This is the IIHF's 2020-2022 bylaw:
https://iihfstorage.blob.core.windo...ads/statutes/2020-2022-statute-and-bylaws.pdf
The relevant part, I believe, is found on the 34th page of the document in section 4.2:
"Players that have never played in an IIHF Championship
4.2.1 A male player who has two legal citizenships or more, or has changed his citizenship or has acquired or surrendered a citizenship and wants to participate for the first time in an IIHF Championship and/or an Olympic competition or in qualifications to these competitions, then in order to play for his Country of choice the player must: (a) prove that he has (i) participated in the competitions within his Country of choice on a consistent basis, over at least 16 consecutive months (480 days) and during two hockey seasons after his 10th birthday (ii) resided in his Country of choice during that period and (iii) neither transferred to another Country nor played ice hockey for a team registered or located within any other Country during that period; and (b) if the Country of his choice is one to which the player has transferred then he must have had an international transfer that was approved by the IIHF and dated at least 16 months (480 days) prior to his proposed participation."
Basically, if my understanding is correct, Lambert would need to play two consecutive seasons of hockey in Canada after he turned 10 in order to be eligible to play for Canada.
This issue came up for Jake Walman a few years back. Born and raised in Canada but he had American citizenship. He was playing NCAA hokey and tried to play for USA at the WJC and the IIHF stopped it.
McKenzie: The curious case of Jake Walman - TSN.ca