Internet not connected after sleep wake even though wifi is connected after awake OS X EI Captain (version 10.11.5).
This issues is consistent, whenever Mac awakes from sleep, My internet is disconnected. I tried the following options,


Turn off wifi and turn back on.
Restart network service by sudo ifconfig em0 down and up.Basically restarting network services.
Network settings to renew DHCP address, Prioritizing the wifi in service order.
I was having the same issue on one of my machines running both macOS Catalina and Big Sur. Tried forgetting the network and joining again, doing an SMC reset, and a few other solutions with no luck.
I was finally able to find a fix for it today (also posted it here https://alihaghani.me/blog/fix-mac-not-having-internet-access-on-wake):

Go to System Preference → Network
Select 'Advanced...' in the bottom right corner
Go to the TCP/IP tab
Enter 'ConnecMac' in the DHCP Client ID field
Select 'Renew DHCP Lease' above the field
Press OK
In the System Preferences → Network, select the Location dropdown
Select 'Edit Locations...'
Select the plus sign to add a new location
Enter 'Home' as its name