first, there's currently no time limit, the game lasts until you feel like not playing anymore.
second, the most reliable source of flowers is the garden room that you can build, check your production building research screen to spend research points to unlock it.
the second most reliable source is the shop. there is a very good chance there's at least one kind of flower on sale every store refresh, though i do usually see 3-5 kinds on average.
there's a chance for petals to be loot when sending servants to the surface to search for treasure, i'm pretty sure.
finally, there's a very small chance loot can be found when excavating a new room, and an even lower chance when a servant enters a room. one of the possible rewards is flowers. i only list this as a possibility, these two methods are not reliable or sustainable.