Single word for describing someone who suffers from a kind of speech impairment

Solution 1:

There is dyslalia:

dyslalia [mass noun] Medicine

Inability to articulate comprehensible speech, especially when associated with the use of private words or sounds.

Origin

Mid 19th century: from dys- ‘difficult’ + Greek lalia speech.

Oxford Living Dictionaries

Solution 2:

In the field of Speech-Language Pathology, speech that cannot be understood by a listener is referred to as unintelligible, which Merriam-Webster defines as

impossible to understand.

There is not a specific word for a person who speaks unintelligibly. In this field, as in many human services fields, there is a preference to use language that puts the person--not the disorder--first, in writing that is meant to be read by the client or family. In casual, non-professional conversation, you could say

She's unintelligible.

or

Her speech is unintelligible.

The second example makes it slightly more clear that the speech itself, not the word choice or grammar, is where the problem is.

Dyslalia (M-W medical dictionary) is a term that was popular in the beginnings of the field of Speech-Language Pathology, but in the last 17 years as an SLP I have never seen the term used except in billing and diagnostic codes, which sometimes hold onto historical language. This Google Ngram shows a spike for dyslalia in the 1940s-50s, followed by declining usage.

There are other terms that are more specific about the origin of the unintelligibility, but they are specialized terms that would be unlikely to be used in casual/non-professional conversation.