The NFLPA has public salary cap numbers for each and currently they are showing that the Kansas City Chiefs have the least amount of salary cap space in the NFL. The Chiefs sit with $2.8 million in cap space while the Detroit Lions are closest with $4.6 million in cap space.
The Chiefs will need a projected $2 million in cap space in order to sign their draft picks, which they'll do some time after the draft on May 2 and sometime before the season in September.
The Chiefs probably aren't going to go into the season with just a few hundred thousand in salary cap space. They're likely going to create some in some way. They can create over $8 million in cap space by restructuring Alex Smith's deal. They can also create cap space by doing something with Eric Berry's contract, although I don't know what options are on the table for that (since we don't know where Berry stands). The Chiefs could still also cut a player in order to create cap space. Only the top 51 players count against the Chiefs cap during the offseason so they're good for the time being. When the season gets closer though, they'll need some space.
GM John Dorsey surely has a plan. He knows the Chiefs roster and their cap situation so I would expect Dorsey to have some sort of plan to create more space before the season.