The problem with that web-calculator is that while it does the modular squaring part (of square-and-multiply) all right, it fails in the multiply part. Apparently the programmer forgot to do those multiplication one at a time, and lazily only tested it with inputs, where this did not result in integer arithmetic overflows, leading to inability to correctly calculate the final remainder.
You can get away with somewhat smaller numbers (still too large for pencil&paper calculations) if you use the Chinese Remainder Theorem. Your modulus factors as
$$63571=151\cdot421,$$ so it suffices to calculate the remainder of $10855^{62807}$ modulo both $151$ and $421$.
Modulo the prime $151$ we need that $62807\equiv107\pmod{150}$, and $10855\equiv134\pmod{151}$. Therefore
$$
10855^{62807}\equiv 134^{107}\equiv 96\pmod{151}.
$$
I didn't check, but there is some hope that your web calculator can manage this exponent with a lower number of bits.
Similarly modulo the prime $421$ we have $10855\equiv330$ and at the exponent we have $62807\equiv227\pmod{420}$. Therefore
$$
10855^{62807}\equiv 330^{227}\equiv 157\pmod{421}.
$$
So, by CRT, the answer is the unique residue class $x$ such that $x\equiv 157\pmod{421}$ and $x\equiv 96\pmod{151}$.
A run of the extended Euclidean algorithm gives that
$$1=33\cdot421-92\cdot151.$$
Therefore
$$
u=33\cdot421=13893\equiv1\pmod{151}
$$
and obviously $u\equiv0\pmod{421}$. Similarly
$$
v=-92\cdot151=-13892\equiv1\pmod{421}
$$
and also $v\equiv0\pmod{151}$. Therefore
$$
x=96u+157v=-847316
$$
has the correct remainders, $96$ and $157$, modulo $151$ and $421$ respectively.
This means that the answer is
$$
x=-847316\equiv42678\pmod{63571}
$$
agreeing with what Mathematica gave for the remainder of $10855^{62807}$ (see my comment under the question).
