Lap Data on User Profile

You can get lap data from users activities in the API, provided they use recording devices capable of creating laps. If you mean something other with laps you have to clarify it, because I don’t know anything about laps in the profile.

It sounds like OP is talking about best efforts - which is a subscription feature - and replicating those outside of Strava is generally frowned upon...

​@arsalwali my 2c, if you want to generate predictions for specific distances, ignore the “best efforts” or anything like that - that won't help to build accurate predictions anyway. What you want is to analyse activities from, say, last 6 months, consider things like consistency, monotony, fatigue, HR, VO2 max, … the most accurate prediction I’ve seen so far is on Runalyze, it's usually very accurate (unlike Strava I’m afraid - those predictions are wildly off I’m afraid).