Emily's quest has many end points depending on the choices you make. It sounds like you reached one of the earlier ends that does not involve acquiring her sister, Tisha, because you had Emily stay with you. The recommended path, the longest one, involves sending Emily away with Tisha and some extra supplies. Emily returns after a few days and then her sister goes missing and you have to search for her.
Assuming that I have the correct interpretation of your game state, there are are no in-game means of continuing her quest. You can either try again during the next play through or edit the save file to revert her questline back to before her sister comes for her.
If you wish to edit the save file, you should make a backup copy(it's hard to fix mistakes) and open it with a text editor (preferably one with a Find or Search function).
Find the entry that looks like:
"emily":6
If the number is anything other than 6, then change it to 6. NOTE: her name appears several times, but you are after the one to the left of the colon.
Find the entry that looks like:
"upcomingevents":[]
It may have other events inside the brackets, those should be left in. You need to add the following value inside the brackets.
{"code":"tishaappearance","duration":1}
If there are other events inside the brackets, be sure to put a comma between this event and those.