I'm not a native speaker of course, but I'd say
宿題は犬に食べられてしまったんです。
Shukudai wa inu ni taberarete shimatta n desu.
This is the passive form of taberu. You often use the passive form when something bad is done to you by someone else. I have even heard it called "meiwaku no ukemi" - "hassle passive".
You can use "ga" (and I think "o" also, according to a Google search) instead of "wa" as the particle for "shukudai", but because you have probably just been asked "where is your homework", I think "wa" is better because it is slightly emphatic.
You use the "te" form of the verb plus "shimau" (here "shimatta" for the past tense) to indicate both that the homework was "eaten up" (shimau indicating finality) and that it was regrettable (which is another use of shimau).
You put in a "n" because you are giving an explanation of what happened.
If you want to "ibaru" (put on an arrogant attitude) you can end with "yo", ("shukudai nara inu ni kuwareta n da yo, monku an no ka?") but normally you would apologize before or after this sentence, or both. Probably "sumimasen" at the start and "moushiwake arimasen" at the end, or something.
I wouldn't bother saying "watashi no inu" because it's probably obvious whose it is. Kids who are of the age to make up such excuses don't usually use "watashi" because it's too formal and grown up anyway, in my experience.