The solution is
21200
This can be found by simply iterating from a starting point. Let's choose 40000 as a start (it's as good as any).
40000 — start with four zeroes (and count them)
40001 — count four zeroes, one four
31001 — count three zeroes, one one, one four
22010 — count two zeroes, two ones, one three
21200 — count two zeroes, one one, two twos
21200 — count two zeroes, one one, two twos
The next row counts the numbers it sees in the previous row. Once we hit two identical numbers back-to-back, we have settled on a solution.
You'll see that 43210, 11111, 22222, 33333, and 44444 are not issues if you make one simple admission.
43210
11111
05000
400001 — allow counting the 5, but don't permanently extend the length
41000
31001
22010
21200
21200