Wolfram|Alpha accepts English language input, not necessarily all types of mathematical expressions.

Mathematica using Wolfram Language, whose capabilities you are implicitly accessing via Wolfram|Alpha, is probably the most efficient tool for your work with arbitary precision numbers, see e.g.

http://reference.wolfram.com/mathematica/tutorial/ArbitraryPrecisionNumbers.htmlthereby also avoiding any ambiguities natural languages have and limitations Wolfram|Alpha needs to impose.

Evaluating your example:

In[2]:= Mod[

9999999999999999999999999999999999999999999999999999999999999999999*\

999999999999999999999999999999999999999999999999999999999999999,

1111111111111111111111111111111111111111111111111111111111111111] \

// Timing

Out[2]= {0.000038, \

111111111111111111111111111111111111111111111111111111111110212}