Yes, LTI does support sending custom data to the tool during launch. For LTI 1.3 (which I assume you're working with, since the older versions are not currently recommended by IMS), the process is as simple as including user-specific data in the 'custom' claim during launch (http://www.imsglobal.org/spec/lti/v1p3/#custom-properties-and-variable-substitution). But how is this done in Moodle?
Moodle provides an LTI subplugin type called 'ltisource'. Among other things, source plugins can augment the data included in launches, by implementing a component callback 'ltisource_PLUGINNAME_before_launch'. This is called when building the launch data to be sent. Given the standard means of including 'extra' data in LTI 1.3 is via the custom claim, you can simple return an item indexed by the name 'custom_YOURFIELDNAME', and this will automatically be added to the claim, and will be available to your tool during launch.
Here's a link to some source plugins (I find reading code easier that reading a forum post!):
https://moodle.org/plugins/browse.php?list=category&id=64
You'll can check out the source code for each plugin on it's plugin listing page.
Hope that helps
Jake