Basically I want to set up a widget / shortcut on their home screen ....
There are a number of social media apps that can do this but they are not one click as you mentioned besides setting up a widget is not possible
Hence, suggesting automation as an alternative
I prefer Macrodroid, being easy to learn and free (upto 5 macros). You would need to install Macrodroid ontheir device
Create a macro
Trigger: Empty Trigger
Actions: Share location
You get the following options - SMS, Twitter , Email , Calendar, Variable
Select SMS and either select contact or number (yours)
Repeat the action above and enter your email after choosing the default e-mail account registered with their device
Constraints : Leave blank
Name the macro and save it
Go to widgets of your home screen and drag and drop Macrodroid widget on any screen. Long press the widget and assign the name of macro (Tip: Install an icon pack that shows visually striking icon so that it is easier for them , like this and replace Macrodroid widget) 
Tell them to press the widget to send you their location by pressing it. Done!
Your SMS would have a link to the location with a pic as below that you can open in Maps. Ditto in your email

Note: GPS needs to be enabled for this to work (obviously). If you can't trust them to remember enabling GPS, that can also be automated. Steps below
For enabling / disabling GPS root permissions are required normally. It is unlikely their device would be rooted, so there is an alternative - adb hack in Macrodroid that overrides this limitation
adb shell pm grant com.arlosoft.macrodroid android.permission.WRITE_SECURE_SETTINGS
followed by:
adb shell pm grant com.arlosoft.macrodroid android.permission.CHANGE_CONFIGURATION
Now enhance your macro (in the actions section)
Location Mode → High Accuracy
Wait 15 seconds ( for location to be registered. May need to tweak it )
Select SMS and either select contact or number (yours)
Repeat the action above and enter your email after choosing the default e-mail account registered with their device
(If you are choosing the email option they would need to enable data but if they forget SMS would go through , so you are covered )
Wait 30 seconds
Location Mode → Location Mode off
This would ensure that battery is not drained by GPS and they don't need to remember to enable/ disable GPS
Yet another approach is mentioned in comments - instead of them doing the work, you do it by adding their device to your account and use apps that track lost devices , choice is yours but I think that the solution mentioned here is superior as it does not call for changes on their device settings and is on demand saving battery
Edit: Another refinement possible , add another trigger in the macro . Missed Call and assign your number to it. When you want their location give a missed call (cut the call before they pick up). You will get the location by SMS. This is handy if they are unreachable for some reason and you want their location. Bonus !!!