Will I have trouble speaking when I get my palate expander?

For a few days. Takes some getting used to. By the end of a week 95% of the speech change will have disappeared, and the last 5% will disappear when expander removed. Make sure you are seeing a qualified orthodontic specialist for this procedure.
Temporaritly. Palatal expander usually causes temporarily change of speech. The most discomfort from the palatal expander would be tongue pain caused by impingement of the tongue during functioning. All these symptoms are usually temporarily.

Related Questions

Help docs! I had my palate expander for 5 years. Will I have to get another palate expander?

Why? I see no earthly reason for wearing an appliance that long. Once the 2 halves of the upper jaw fuse there is no way to correct transverse skeletal discrepancy without surgery. Please seek consultation with one or more qualified orthodontic specialists rather than with well meaning but under-trained dentists who dabble in "tooth movement" which is not synonymous with "orthodontic treatment.". Read more...
Too long. A palette expander is typically worn for weeks, not years and is typically not used too many years into adulthood. Read more...