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}