Cloud camera solution
TL;DR Use SeeCiTV to use a second device as a remote camera.
Assuming your device is always going to be plugged in and connected to the Internet...
You can use the app SeeCiTV on the Play Store in order to always be able to check on the camera on your device. There are several apps which accomplish this, but I recommend this app as its interface seems easy to use (unlike the IP Webcam solution posted below).
This method is secured using your Google account which should already be set up with your phones, with the addition of being able to add Trusted Users if you want to give other accounts access to that camera.
One potential downside to this is that you can only access your new "home camera" on another phone or tablet, as there is currently no way to view it from a PC (unless you have a mobile device emulator). While SeeCiTV is great in terms of security and efficiency (the device screen and camera is turned off until you want to monitor it), let's say you need to be able to access it from any device on the Internet, then you'd need a different app like the next solution...
This is another handy solution because you will not need to keep your remote screen on at all times (unlike the next few solutions). This supports most devices, but you can check the supported manufacturers list to make sure your device will be supported.
Download and install TeamViewer QuickSupport on the Play Store and set up a TeamViewer account from the app. Now download the TeamViewer control app on the Play Store on both the remote device (the device you will control) and the host device (the device from which you will he controlling your remote device).
Now you will need to install a manufacturer-specific add-in for QuickSupport, which the app should prompt you to install when you first open it. After you install the add-in, sign into your TeamViewer account on both devices. Then, add your remote device's ID (given by the QuickSupport app) to your list of account devices. Then, you'll be able to connect to that device any time and answer any incoming calls that you need (or you could start the call from the remote device).
IP Webcam solution
TL;DR Please use the Cloud camera solution unless you know what you're doing here.
If you need to have access to this device from a PC, and you can't use a mobile device or desktop emulator, then you can use an app like IP Webcam on the Play Store in order to constantly stream the camera data to the Internet.
You'd have to set up your remote network's ports and other settings to work with IP Webcam. The steps are walked through in the app, but
Why do I recommend the Cloud camera solution instead of this one?
For starters, it's potentially much less secure. SeeCiTV uses a Google account for authentication, but IP Webcam has its own form of authentication. Since you'd be connecting to your own remote IP, it's possible that your credentials, when you log in, will be transferred insecurely over the network. While IP Webcam may encrypt this information before transferring, you can avoid exchanging your camera stream and passwords over a non-SSL-secured connection if you use the Cloud camera solution instead. Even if you use a different password for this, it is much easier for attackers be able to gain access to your remote camera. And, yes, they could watch your baby for you, but maybe you should leave that task to yourself and your family.
It's also far less convenient to set up and to access the IP Webcam. IP Webcam will require your screen and camera to be active at all times (which will affect device life).
TL;DR If you need 2-way comms, use Skype
If it's necessary to have two-way communication between your device and the remote device, you could set up a Skype group between 3 devices: the remote tablet, a PC or another old device, and the device you want to use to video call. First, you'll need to install Skype on all of the devices. You can use these installers: Skype for Android or Skype for desktop. Start a Skype group between the devices, using a different Skype account for each device. Start a group call, and you will need to answer on the tablet and PC/device, then start the video on the remote tablet. Now you can connect to the group call whenever you want, from a third device. You can even share your own video feed from your third device. Just make sure your Internet is stable and that the call remains on at least 2 devices at all times.