Yes, your SEO firm should be involved right from the start. Even better, your website redesign AND your SEO firm could be the same company, like Social Power!
Redesigning your website without including SEO along the way is a big mistake for several reasons:
Most website developers are not SEO professionals, and as such tend to omit much of the SEO work already done on your behalf. This means your SEO manager will have to go in after the fact to fix everything that has been missed.
Website redesigns tend to include new web pages and remove old ones. This looks like missing or broken pages to Google when the new website is launched. All the SEO juice associated with those pages disappears. This is a double whammy for the price of one oversight.
Webmasters will typically not use SEO optimized URL/folder structure that was built by the SEO firm.
Keyword optimized URLs are often replaced by geeky tech sounding URLs.
H1 and H2 title tags are often replaced with other tags that are not SEO optimized
Internal linking (a lengthy and time consuming process to develop) is usually forgotten and ignored.
Content that is SEO rich and designed to build trust and relevance is replaced with copy from the marketing department.
Business owners and managers tend to try and save money by pausing their monthly retainer with their SEO partner during the website redesign. This comes at a time when they need the SEO perspective more than ever! In the end it will cost more to employ the SEO professional to start over with the on-site attributes and clean up the mess.