How many polynomials are used in Secure Multiparty Computation?
For "standard" Shamir secret sharing you use a new random polynomial for each secret you want to share (including when you use Shamir secret sharing in SMC). I.e., you would use different polynomials for each secret. Note, that Shamir secret sharing does not directly work on "words" but rather elements of some field. You would have to translate your data into field elements and then share those elements individually. 

There is, however, a variant of Shamir secret sharing called "packed secret sharing". In this variant you are able to share a "block" of $l$ secrets using a single polynomial. This means you would be using a factor $l$ less polynomials. There are SMC schemes that rely on this kind of sharing as well, although they are somewhat more complicated than those using standard Shamir secret sharing (an example of such a scheme can be found here: iacr eprint link).

As to how search would work, this is really not something that is standardized. The SMC allows you to compute any arithmetic circuit over the shared data. You can essentially express any function, including keyword search, as an arithmetic circuit. However, exactly how to do this is not trivial to figure out.

In fact, for this particular task you may be better off picking a different SMC technique, based on Boolean circuits instead of arithmetic circuits. This is because keyword search probably involves a lot of comparisons. Something that Boolean circuits are much more well suited for.