AGPL is a "viral" or "strong copyleft" license [0], like GPL: if any part of a piece of software is AGPL, the entire software must be AGPL.
The license [1] says this in section 5:
> 5. Conveying Modified Source Versions.
> You may convey a work based on the Program, or the modifications to produce it from the Program, in the form of source code under the terms of section 4, provided that you also meet all of these conditions:
> [...]
> c) You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. This License will therefore apply, along with any applicable section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged. This License gives no permission to license the work in any other way, but it does not invalidate such permission if you have separately received it.
If you intended it to be non-viral (weak copyleft) and based on distributing modified versions of your files, the Mozilla Public License [2] may be a better fit for the frontend libraries. Then people could use them unmodified in software with any license, but they would have to release source code of modified versions of those files. This would not do what you want for the server software though, as the "release source code" only applies when distributing software (which is not happening if it's staying on the server).
I am not an expert, and it is not clear to me whether the frontend and backend constitute a single work according to the license. This is important to investigate. If they did count as a single work, rather than two separate works communicating, then the AGPL from the server software would apply to the client software and licensing the frontend libraries MPL wouldn't work. This says something about that in the other direction: https://www.gnu.org/licenses/gpl-faq.html#AGPLv3ServerAsUser
The license [1] says this in section 5:
> 5. Conveying Modified Source Versions.
> You may convey a work based on the Program, or the modifications to produce it from the Program, in the form of source code under the terms of section 4, provided that you also meet all of these conditions:
> [...]
> c) You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. This License will therefore apply, along with any applicable section 7 additional terms, to the whole of the work, and all its parts, regardless of how they are packaged. This License gives no permission to license the work in any other way, but it does not invalidate such permission if you have separately received it.
If you intended it to be non-viral (weak copyleft) and based on distributing modified versions of your files, the Mozilla Public License [2] may be a better fit for the frontend libraries. Then people could use them unmodified in software with any license, but they would have to release source code of modified versions of those files. This would not do what you want for the server software though, as the "release source code" only applies when distributing software (which is not happening if it's staying on the server).
I am not an expert, and it is not clear to me whether the frontend and backend constitute a single work according to the license. This is important to investigate. If they did count as a single work, rather than two separate works communicating, then the AGPL from the server software would apply to the client software and licensing the frontend libraries MPL wouldn't work. This says something about that in the other direction: https://www.gnu.org/licenses/gpl-faq.html#AGPLv3ServerAsUser
[0] https://en.wikipedia.org/wiki/Copyleft#Strong_and_weak_copyl...
[1] https://www.gnu.org/licenses/agpl-3.0.html#license-text
[2] https://en.wikipedia.org/wiki/Mozilla_Public_License