This app was mentioned in 6 comments, with an average of 1.00 upvote
I use app for taxi called FindTaxi: https://play.google.com/store/apps/details?id=org.sleepnova.android.taxi&hl=en&gl=US
I prefer it over Uber, as the local taxi drivers can be interesting characters.
Most should take credit card, Easycard, or Line... but that doesn't mean there aren't drivers that will ask for cash only.
I use an app called FindTaxi, they have an option to sort by credit card and it allows you to read your drivers profile before contacting them. https://play.google.com/store/apps/details?id=org.sleepnova.android.taxi
I haven't had any of those issues with taxi's in Taipei... My advice is try one of the taxi apps such as "Find Taxi". It shows the taxi drivers around you on a map, their ratings, and let's you call/text them ahead of time (they often have LineID in their profile). You can even request things such as English driver, child seat, etc.
Here is a link to the Android app: https://play.google.com/store/apps/details?id=org.sleepnova.android.taxi&hl=en_US&gl=US
Taxi's are everywhere and way more reliable than Uber. You should be able to find a van if you need it... but I'm not sure about the legality of kids and car seats.
I use the FindTaxi app on Android, it has a pretty good search function that let's you look for English speaking drivers, vans, car seats, etc etc etc. You use the app to find the taxi, and then you pay the fare as you normally would.
Not used it but this suggests it may work..
https://play.google.com/store/apps/details?id=org.sleepnova.android.taxi