If you transform your main website’s WordPress into a multi-site environment, you can move blog.yourdomain.com to be a network site.
Once you do this, you could use the Multisite Global Search plugin to add a widget that searches all network sites.
It would be easy to move your standalone blog.yourwebsite.com to a network site:
- export the content from blog.yourdomain.com,
- create a subdomain WordPress network on your main website,
- edit your blog.subdomain.com settings (Settings > General) to change the site and WordPress URL to blog2.subdomain.com,
- rename your subdomain blog to blog2 in cPanel,
- create a network site called blog.yourdomain.com
- import the content you exported earlier.
Done.
PS – You’d need to copy any themes and plugins you have on blog2.yourdomain.com into the main WordPress installation.