Good morning Rotsen!
An excellent framework to create applications in PHP is www.symfony.es
However, you could also choose to develop within the Microsoft .Net framework (in ASP.Net). It is usually very suitable to start with development and efficient for small applications.
I do not know the level of expertise you have in software development, but keep in mind that you usually study several years to program applications. It is essential to have a knowledge base on how computers work and then on the basis of algorithmics.
Regarding the topic of hosting , the answer is yes. You could host it yourself on a local server in the company. However, this is not related to the fact that it will only be used by a few users, since it depends on the computer architecture that you will limit the users that it supports. I mean there are large data centers with very powerful servers that are used locally. In turn, there are hosting with different types of computational power that can accommodate different "sizes" of applications. With all this I'm going to first you should see what computers you have to mount the server. I always recommend having a dedicated computer to be a server and not mount the database and the web server on a computer used by users.
Finally, hosting it yourself in the company could also make it "online". Surely you mean that it can be accessed from the Internet, so for that you could forwarde the port on the company's router to the web server and get a free dynamic pointer and thus access the application from anywhere. I mean it is not a necessary condition that you hire a hosting to access the application from anywhere.