I am not entirely new to Linux but I have no experience in configuring Linux bridgets / routers.
Recently I installed Ubuntu 16.04 on a Dell Optiplex 990.
It has two network cards and one of these is a VPN client installed.

My network contents:
- modem/router 192.168.40.1 DHCP range 100-200
- Laptop (DHCP)
- PC (DHCP)
- Rasberry PI3 192.168.40.36
- Mediaplayer (Linux) 192.168.40.25
- Nas 192,168.40.10
Shares accessabel by Laptop, PC, Mediaplayer
- Linux machine (Ubuntu 16.04)
NIC 1 192.168.40.40
NIC 2 192.168.40.41
VPN-client installed on NIC 2
(I made a drawing but I do not know how to upload it)

The intention now is that other machines, Mediaplayer and PI3, in the network can use this VPN client.
It remains necessary for the media player to acces the share.

Is this possible and how do I do this (which steps?)?

Where are you trying to connect exactly? What kind of VPN? Where would the VPN server be? You've only talked about the VPN client, but are you going to configure the VPN server or do you want to buy it?

Thanks for the reply.
My first test was to install the VPN client on my modem / router.
That was successful. Only the speed of connection was too slow.
As an alternative, I want to use the linux machine ((with the two network cards, neither of which are wireless).
All mentioned machines are in the home network.
The PC and laptop must access the internet via the router, without VPN.

The VPN connection is rented from a VPN provider.

In the original post I tried to upload the network design.

The machine with VPN is the client. Usually, if you want different devices to use the vpn, you put a client on each one, and use the same credentials.

You do either what AwesomeMachine says, or you use the linux machine as a router for the other machines in your network and add a permanent static vpn route. The problem with this setup is that you'd end up with nat within nat, which isn't exactly what you'd want, although it's possible.

By the way, you don't install a vpn client on a NIC, you install it on a machine and then you set up the routes. It's improper to say that a vpn client is installed on a certain NIC.