Question

What Pokemon obtainable in X can learn Cut, Strength, Waterfall, and Rock Smash?

Comments

For future reference, there is a nifty tool on this site that will let you put in moves and search for a Pokemon able to learn all of them, though it doesn't allow you to filter by generation.  There is also a reference page linked there for Pokemon that have been qualified as HM Slaves.

http://pokemondb.net/tools/moveset-search

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^^ I had no idea that existed. Sometimes this site never fails to surprise me. o.o

Answer 1

The following Pokemon can learn Cut, Strength, Waterfall and Rock Smash in X and Y:

• Dragonite
• Mew
• Croconaw
• Feraligatr
• Corphish
• Crawdaunt
• Prinplup
• Empoleon
• Bibarel
• Arceus
• Samurott
• Froakie
• Frogadier
• Greninja

Pokemon in bold are easily available in X and Y.

Hope I helped. :)
Source: The respective Serebii Attackdex pages for the four moves + some Excel magic

Answer 2

A Bibarel will do the job.

He can learn Cut, Rock Smash, Strength, and Waterfall

You can find one Bidoof on route 3 and Route 22.