This will be a speculative answer because the sum total of information about Tony Stark’s association was already given in your question without supporting details ever having been revealed:
"You have reached the Life-Model Decoy of Tony Stark."
Tony Stark obviously knows about LMD’s. So how?
The program is not new.
We learn in Agents of Shield, Season 4 Episode 12 “Hot Potato Soup,” that Billy and Sam Koenig participated in the original Life-Model Decoy Program of S.H.I.E.L.D. as programmers and technicians.
As the Wiki notes:
"However, despite their efforts, they never managed to create a fully functional android, so the project was eventually abandoned."
No doubt Stark, with Stark Enterprises' long association with SHIELD affairs, gained knowledge of that program’s history. Speculating any more than that such as how he came by that knowledge would be fan fiction at this time.