Exist algorithms without input data?
Yes, of course they exist, and they're useful both practically and theoretically.

Any time you write a computer program to solve some specific task that's hard-coded into the program (for example, to compute some value or find solutions to a specific set of equations you're interested in), that's an algorithm with no input.

If you want to prove undecidability of, e.g., the problem of determining whether a Turing machine halts for every input, the first step is to translate a program that uses its input to a program that doesn't use its input and always computes as if it received some specific input. That's another example of an algorithm without an input.