I need to have security certificates renewed soon on several RedHat servers. Will I need root access to install them?
TL;DR: Probably yes, but if you don't even know how to tell, you will probably need somebody more experienced who knows your particular system to help with this.
It depends where they're installed, and the permissions on the file in question. Certificates (and their corresponding private keys) are usually stored in files, like any other data, and usually stored in a base64-encoded or binary format. For web servers in particular, there's usually a config file specifying the location of the cert and key (though the location of that config file, and the cert+key files it points to, will vary depending on things like server software, version, installation source, choices made at install time, and choices made since installation). Like all other files, those files will have access controls (classic Unix file permissions and/or full ACLs), which will control who can edit them.
It is common with development servers (those not intended for production use) for the user to have the ability to edit the server config directly, as this is commonly needed during development anyhow. However, when deploying a server in production, it's common to create a unique user account for it (which you, the user, cannot log in as even if you can log into the server at all), and it's a good security practice to further lock down the server's static files (very much including the config file and cert + key) so that even the server can't modify them; this is usually done by restricting write access to root only. So yes, you'll probably need root since these are probably production servers. But there's no way I can tell just from the info you've given us.
RedHat is developer of GNU/Linux distributions, and the distros they publish have many variations across versions. Furthermore, RedHat distros can run all of the huge range of Linux server software (which may come from official repos, unofficial repos, direct downloads from the developers, official source downloads compiled and installed manually, source downloads manually patched and installed with custom paths... and of course each comes in many different versions which have different configurations and defaults). Even restricting to some likely defaults (one of the most recent versions of RHEL, running one of the more popular web server packages from the official RedHat repos, taking default install options) there are too many possibilities for me to state confidently.
On the other hand, if you know where the server's configuration files are located, a simple
ls -lA on the relevant directory will probably tell you what permissions are needed to modify the relevant files.