Ask your DM*
* And hope they don't have a bias against amphibians
As the Grung is not an officially released race, it is not clear how it interacts with class features and may not be fully baked. While you can look at the rules to try and parse how this works, the fact that we don't fully understand the initial poison mechanics of the grung makes it hard to give a rules-based answer as to whether or not it would work. A lot of this is because the Grung race itself seems like it's more homebrew - but the real issue is that the race doesn't describe how that mechanic works fully enough to rule on (which may be an issue stemming from it's creation.)
The following is from its description on DMs Guild (emphasis mine):
Summaries of the characters from the new Twitch streamed game in all their glory, plus the officially UNOFFICIAL grung traits used to created them...
Disclaimer: Outside of a certain surrogate character, this document in no way makes grung a legal, playable race in the D&D Adventurers League, or any other campaign where the Dungeon Master hates amphibians with an excess of apostrophes in their names… and yes if you’re not immune to poison and you must touch the grung to heal it… you get to make a saving throw.
If you're the DM, you should consider in deciding as to whether or not the thing that makes the Grung poisonous is available in its wild shape form. Not necessarily how it delivers it, but how it produces the poison.
If you and your DM thinks it makes sense, then roll with it. If not, then don't.
The race is from a twitch event
The Grung was created for a twitch event by WoTC and was released as a supplement to Tomb of Annihilation whose sales would benefit ExtraLife. They seem to go out of their to state this is far from official, but in a very tongue-in-cheek way.
The creation of the race seems to have been for this event and their clarification on it's level of officialness and lack of detail makes it difficult to review the race in terms of currently existing rules. Because this sits outside the official rules, it will very much be up to the DM to decide how and if this interaction will work (and if they'll allow this race at all.)