I am using the Wordpress REST API to provide a back end to my React JS front end. Right now both are running on separate Apache servers on AWS and all works well.
Is it possible to run both on the same server? So that my React front end is making requests to Wordpress on the same server?
Sure, if you want to use Apache to handle distributing your frontend and handling the WP backend, read up on Virtual Hosts: https://www.digitalocean.com/community/tutorials/how-to-set-up-apache-virtual-hosts-on-ubuntu-14-04-lts
The gist of it is that you'll have a VH for your frontend
<VirtualHost *:80>
ServerAdmin admin@example.com
ServerName example.com
ServerAlias www.example.com <-- see here
DocumentRoot /var/www/example.com/client
ErrorLog ${APACHE_LOG_DIR}/error.log
CustomLog ${APACHE_LOG_DIR}/access.log combined
</VirtualHost>
and a VH for your backend
<VirtualHost *:80>
ServerAdmin admin@example.com
ServerName example.com
ServerAlias api.example.com <-- and here
DocumentRoot /var/www/example.com/server
ErrorLog ${APACHE_LOG_DIR}/error.log
CustomLog ${APACHE_LOG_DIR}/access.log combined
</VirtualHost>