Setting up a site with private infomation without a password?

I have multiple employees that have iphones. We all access the same website from our devices. Currently I have it set up with username and password but would like to eliminate the need to keep typing in the password every single time. Is there a way for the website cookies or something to remember the individual phone and stay logged in or something?

X.509 server and client certificates would accomplish this, and give you SSL (HTTPS) encryption as well. The steps for setup are specific to whatever webserver/clients you are using, but here's a general outline:

The server cert uniquely identifies the webserver, so the clients know that they are indeed talking to the correct server, not a man-in-the-middle attack.

The client certs uniquely identify each client to the server, again to avoid MITM. And together the server/client certs also serve to set up the public-key/private-key encryption for the connections.

I'd setup a mobile login page that requres a 4 digit PIN or something and then drops a cookie on the device. I'm not sure how deep your knowlege goes, but you could also register each devices ECID or UDID with the site so that the device itself has access to the site.
The problem is that doing something like this leaves a huge vulnerability in your process if someone loses their phone.
That's why I'd do a separate mobile login page that requires a PIN - using 4 digits as a PIN gives 10, 000 different possibilities so it's not like you will run out of original possibilities. Allowing for letters would make it up in the billions of combinations too - so it'd be might y secure. This will also make it easy to forgoe the cookies idea completely and keep everything private.