These should be rearranged a bit to follow the PEP-8 format of standard lib, third-party, this package. You can also catch ImportError without assigning it to a variable because you don't use the exception.

import logging

from django import forms
from django.utils.translation import ugettext_lazy as _
try:
    from jira.client import JRIA as JIRAClient
    from jira.exceptions import JIRAError
    has_jira = True
except ImportError:
    has_jira = False

from reviewboard.hostingsvcs.bugtracker import BugTracker
...

Indentation in here has an extra 4 spaces.

This could use self._get_issue()

When logging things from within an exception handler, it's usually nice to add exc_info=1 to the logging.*() parameters.

This could use self._get_issue()

Should add exc_info=1

Should add exc_info=1

Is it worth caching the result of this call anywhere? It seems likely that if you want one part of the issue data, you'll want the others.

You may want to also trim whitespace.

Please add a docstring.

Please add a blank line between these two.

These could be combined into one statement.

Please add a docstring.

Please add a blank line between these two.

These can be combined.

Please add a docstring.

Please add a blank line.

These can be combined.

Can you reformat this to put all the keys one indentation level down?

issue = {
   'description': ...,
   'summary': ...
   ...
}

This should probably also include the URL of the JIRA server, in case there are multiple JIRA servers present.

These could be combined into a single statement.

If we get an exception, this will return None, which your other change will then just insert into the template. Should this return ''?

Same question about None vs ''.

Same question about None vs ''.