Interesting question! Here's some thoughts I have on it as a research assistant in a developmental psychology research lab with a degree in psychology, however by no means an expert so I expect the topic is much broader than I'm able to cover here.

Technically speaking, there is no official diagnosis named dyslexia. It is subcategorised as a distinct type of Specific Learning Disorder in DSM-V (the most current diagnostic manual), characterised by deficits in different aspects of reading and writing. Other subcategories are characterised by, for example, difficulty with arithmetic skills, number sense and mathematical reasoning (or what one might call dyscalculia). I'll call this subcategory of specific learning disorders focused on reading difficulties dyslexia for simplicity in this case.

According to the criteria for dyslexia, an individual should have difficulties with various aspects of reading and writing, and these difficulties should have a significant impact on academic achievement, usually identified through standardized testing. So, in a society without written language (and hence no standardized testing) it's not strictly speaking possible for an individual to have dyslexia, because the conditions for diagnosing the disorder as it is defined today did not exist.

However, if you delve more into the origins of dyslexia, the answer isn't quite so clear. Research into the etiology (origin/cause) of dyslexia has found consistent differences in the neuropsychological activity between children with dyslexia and children without during a word recognition task, suggesting that dyslexia or at least the underlying cause of it didn't appear after the development of written language, but have a physical component. Research also suggests that there may be a causal relationship between phonological processing (i.e. the use of sounds in a language) and dyslexia - deficits in phonological processing may precede reading difficulty. Phonological processing includes awareness of the sound structure of a language, like being able to hear a word and break it down into its constituent parts.

One possible interpretation is that this difficulty in processing the sounds used in language leads to difficulties with reading and writing, due to an impaired ability to break a spoken word down into separate phonemes and translate those phonemes into corresponding written letters and vice versa. Symptoms of dyslexia are extensive and can vary significantly between individuals, and it is occasionally possible to spot some symptoms in pre-school children, which can include delayed speech development, speech problems (like with pronouncing long words or swapping letters), problems expressing themselves verbally, and difficulty remembering words or using incorrect words in a sentence. These things are quite vague and can have many different causes, or have no cause at all beyond some kids simply being slower learners, and as such are not diagnostic criteria on their own.

While an individual in a society without written language wouldn't have had dyslexia as we define it today, it is entirely conceivable that individuals with deficits in phonological processing may have existed and may have shown 'symptoms' like having difficulties with speech, or that someone who would today be diagnosed with dyslexia but doesn't display the same difficulties with phonological processing wouldn't have experienced any noticeable differences.

As a footnote, dyslexia is often comorbid with other developmental disorders - people with dyslexia are statistically more likely to also have dyspraxia, a disorder that affects development of fine motor coordination skills. I imagine something like that would be much more noticeable in a 'caveman' society!