Issues with Multisite installation and 403 error for REST API request

If you’re using WordPress Multisite in subdomain mode, here’s the recommended .htaccess file WordPress section (see the WordPress support page):

RewriteEngine On
RewriteBase /
RewriteRule ^index\.php$ - [L]

# add a trailing slash to /wp-admin
RewriteRule ^wp-admin$ wp-admin/ [R=301,L]

RewriteCond %{REQUEST_FILENAME} -f [OR]
RewriteCond %{REQUEST_FILENAME} -d
RewriteRule ^ - [L]
RewriteRule ^(wp-(content|admin|includes).*) $1 [L]
RewriteRule ^(.*\.php)$ $1 [L]
RewriteRule . index.php [L]

(Assuming that you started with WordPress version 3.5 or higher. For WP 3.4 and lower, see the support page linked above.)

Your .htaccess file looks like it’s actually using code from the subfolder Multisite installation.