Is there a way since (iOS 7’s release) to get the UDID without using iTunes on a PC/Mac?

Navigate to http://get.udid.io/ from Safari on your iOS device. It works like a charm and requires neither iTunes nor any other computer. No app installed either.

EDIT:

Also, have a look at Getting a device UDID from .mobileconfig if you (understandably) would rather have this .mobileconfig certificate hosted on a server of yours.

MAKE YOUR OWN:

Have a copy of the .mobileconfig example hosted on your server and write 2-3 small scripts in your favorite language to handle the following flow:

1. Navigate on Safari to a URL redirecting to enroll.mobileconfig below. This makes iOS open the Settings app and show the profile.
2. Upon accepting the profile, iOS switches back to Safari and posts the DeviceAttributes array to the URL specified enroll.mobileconfig.
3. The POST data will contain a .plist file with the requested attributes (see example below). One of which will be the holy UDID.

Remark: You should probably have some user friendly messages. Specifically, we even have a step 0. where the user is asked to provide their name and e-mail that we store temporarily in the HTTP session and then redirect the request to the mobileconfig profile. We ultimately match this info with the iPhone data and send a friendly confirmation e-mail. HTH.

enroll.mobileconfig

<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd">
<plist version="1.0">
    <dict>
        <key>PayloadContent</key>
        <dict>
            <key>URL</key>
            <string>http://support.devcorp.com/uuid/returnurl/</string>
            <key>DeviceAttributes</key>
            <array>
            <string>DEVICE_NAME</string>
            <string>UDID</string>
            <string>PRODUCT</string>
            <string>VERSION</string>
            <string>SERIAL</string>
            </array>
        </dict>
        <key>PayloadOrganization</key>
        <string>DevCorp Inc.</string>
        <key>PayloadDisplayName</key>
        <string>Profile Service</string>
        <key>PayloadVersion</key>
        <integer>1</integer>
        <key>PayloadUUID</key>
        <string>C5FB9D0D-0BE7-4F98-82CC-5D0EA74F8CF8</string> <!-- any random UUID -->
        <key>PayloadIdentifier</key>
        <string>com.devcorp.profile-service</string>
        <key>PayloadDescription</key>
        <string>This is a temporary profile to enroll your device for ad-hoc app distribution</string>
        <key>PayloadType</key>
        <string>Profile Service</string>
    </dict>
</plist>

sample .plist POSTed by the iPhone to the given URL

<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd">
<plist version="1.0">
<dict>
    <key>PRODUCT</key>
    <string>iPhone4,1</string>
    <key>SERIAL</key>
    <string>DNPGWR2VCTC0</string>
    <key>UDID</key>
    <string>b01ea7bc2237fed21bfe403c6d2b942ddb3c12c3</string>
    <key>VERSION</key>
    <string>11A465</string>
</dict>