Is it possible to design a Turing Machine without extra symbols for this language?
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