Can I advertise my Clarity service on social media? Is there a plug-in for sharing on my website?

I think you are looking for a small button to go alongside Facebook and LinkedIn? Great idea!
While this is not an official answer, I would think you could create a button that links to your clarity profile and add it to your site, ie, "Find me on Clarity," depending upon your technical expertise.

To optimize your listing for Clarity's internal search, as well as for major search engines like Google and Bing, make sure to do the following:
1. Include potential search terms in your title and description.
Think about how people search for expertise. Help with Social Media marketing might include words like Twitter, Google+, Social Strategy, etc. Mention any companies or past clients that you've worked with. Include detailed descriptions of your accomplishments and related tools & techniques you might use.
2. Set a competitive hourly rate.
Look at what other experts in your category are charging and use a similar price. It’s a good idea as a new expert to start off at a slightly lower rate, then raise it as you get more reviews. Searches are made using specific keywords and quality ratings. Making sure your profile details are up to date will ensure you are reaching the right members.
3. Complete each section of your listing and your Clarity profile.
Utilize all of the fields available to manage your listing and profile. Be as descriptive as possible as to your accomplishments, background, and expertise. High quality photos are extremely important.
4. Use social media and the Web.
Make sure to "tweet", "like" and "+1" your profile, and don't be shy asking your friends to help! Get your listing's URL from the address bar on your public page. If you or your friends have websites, be sure to provide the link there. Relevant links to your listing from other websites can help you rank higher in major search engines.
5. Favourite your profile.
Favourites are a simple way to save and share profiles with colleagues and friends. Be sure to have your friends favourite your profile and listings and share them publicly through Twitter and Facebook.
6. Add your blog widget to your website.
Every expert has a blog widget that they can add to their personal or company website. Doing so will promote your profile to any of your visitors and encourage them to request a call.
7. Use your VIP link to build your reputation.
Clarity provides all experts with a VIP link they can use to make their profile FREE. Using Clarity to take advisory or coaching calls with existing clients will help build your reputation and search rankings.
8. Answer questions to demonstrate your expertise.
Clarity provides a private way for members to ask experts questions. Experts can build their reputation and increase demand for their time by providing great answers. Members typically follow up with a call, and all answers are listed on your profile for future visitors to search.
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath

You can pretty far by just copying the URL link from your profile page and using it stand along or embedding as the action in images, buttons etc

CLICK HERE: https://clarity.fm/bobschwartz

I don't think we have that option anymore. Back in the day when we started we used to have a mobile app as well but over time to streamline efficiencies some items have been removed over time, including plug ins and banner codes..