Determine if $x^3+y^3+z^3+t^3 = 10^{2021}$ has a solution

Easy, notice that $10^{2021}=100\times 10^{3\times 673}$. Next use your code, but for the factor 100.

FindInstance[{x^3 + y^3 + z^3 + t^3 == 100, 0<x<y<z<t}, {x,y,z,t}, Integers]

yielding a single result

(*{{x -> 1, y -> 2, z -> 3, t -> 4}}*)

Now verify the solution

(x^3 + y^3 + z^3 + t^3 /. {x -> 1 10^673,y -> 2 10^673,z -> 3 10^673,t -> 4 10^673}) == 10^2021
(* True*)