Is it possible to design a Turing Machine without extra symbols for this language? Is it possible to design a Turing Machine for the language defined as L = {0n1n | n >= 0} with only the symbols in the set of {blank, 0, 1}?
Yes, listing out all the transitions would be tedious so I'll just give you the high level algorithm (the idea is to iteratively match the first 0 with the last 1, while deleting both of them). Starting from the first symbol on the tape do the following:


If the current symbol is $\varepsilon$: accept
If the current symbol is $1$: reject
Delete the current symbol (by writing $\varepsilon$)
Move right until you find the next $\varepsilon$ symbol
Move left (the head is now on the last non-blank symbol, if any)
If the current symbol is $\varepsilon$: reject
If the current symbol is $0$: reject
Delete the current symbol (by writing $\varepsilon$)
Move left until you find the previous $\varepsilon$ symbol
Move right (the head is now on the first non-blank symbol, if any)
Repeat