SEO audit is perhaps the first stage of a SEO process. You may not even have access to the Google Analytics and Webmaster tools account but still need to make recommendations and sketch out an initial strategy plan.
An exhaustive SEO audit essentially involves an analysis of the current stage of the website, which at least includes keywords, duplicate content issues, duplicate Meta tags, rankings, pages indexed and a lot more.
Following are some FREE SEO Audit tools that can help you in performing a comprehensive SEO audit. My personal favorites are the XENU Link Sleuth and SEM Rush.
1) XENU Link Sleuth: Running the URL in XENU will give you a very good insight into the overall website architecture. Discover, titles, Meta tags, check duplicities, redirects, file types etc. The best way to use this tool is to extract the above data in an Excel and do further analysis.
2) SEM Rush is extremely helpful in gaining insights into keywords. You also get a brief overview on who the competitors are. A more detailed explanation on SEM Rush is here. www.semrush.com
3) SEO MOZ Open Site Explorer: This is more of a link analysis tool. While you may to have access to Google or Bing Webmaster console, you can still analyze the overall link profile and popularity of the website with Open Site Explorer. moz.com/researchtools/ose
4) Ahrefs lets you analyze the backlinks, keywords, anchor texts, top content, referrers, and broken links. You can also export the data in PDF and CSV format. ahrefs.com
5) Screaming Frog: The FREE version of screaming frog SEO spider tool lets you find broken links, errors, redirects, title tags, meta tags, meta robots, duplicate pages. However, it comes with a crawl limit of 500 URLs. The paid version does not have a URL crawl limit. www.screamingfrog.co.uk/seo-spider/
6) WooRank generates an SEO score for the website. The FREE version is quite exhaustive. SEO audit information includes Title tags, Meta tags, keywords, alt attributes, indexed pages, robots.txt, xml sitemaps, backlinks, social etc. The 2 features that stands out is keyword consistency, mobile performance and usability. www.woorank.com
7) Raven Tools (raventools.com/site-auditor/): FREE but requires sign up. You can also login with your Google account.
8) Web Confs lets you analyze a domain or a keyword. Shows limited information on keywords and rankings. www.webconfs.com/free-seo-audit-tool.php
9) Deep Crawl (www.deepcrawl.com): Overview on site architecture, competitors, possible penalties. Has a free as well as paid version.
10) SEO Optimizer (www.seoptimer.com) : Parameters include, H1-H6 tags, meta tags, title tags, links, content and robots.txt
11) Alexa offers a free trial version. This gives you a good view of keyword competitiveness and performance, traffic, engagement and reputation metrics.
Hope this helps. Do let me know if I have missed out on any tool that can help in auditing the SEO credibility of a website.