Yes, you can, but you have to modify the headers in the email_*.tpl template files.
This is the existing Invitation template.
CODE
To: {$user.email}
Subject: Invitation to Participate in Survey
<!-- HEADER SEPERATOR - DO NOT REMOVE -->
Hello {$user.name}. You have been selected to participate in
a survey at the following site. You will need the invitation
code listed in order to access the survey.
Survey: {$survey.name}
Invitation Code: {$user.code}
The following URL already contains your Invitation Code, so
clicking on it or typing it into your browser will take
you directly to the survey.
{$user.take_url}
{section name="results" loop=1 show=$user.results_priv}
You can view the results of this survey at the following URL.
{$survey.results_url}
{/section}
Or, you can alternatively find the survey from our Main
Page and provide your Invitation Code when prompted. The
following URL will take you to our Main Page.
{$survey.main_url}
Try modifying it to this for an HTML only email.
CODE
To: {$user.email}
Subject: Invitation to Participate in Survey
Content-type: text/html
<!-- HEADER SEPERATOR - DO NOT REMOVE -->
<p>Hello {$user.name}. You have been selected to participate in
a survey at the following site. You will need the invitation
code listed in order to access the survey.</p>
<p>Survey: {$survey.name}
Invitation Code: {$user.code}</p>
<p>The following URL already contains your Invitation Code, so
clicking on it or typing it into your browser will take
you directly to the survey.<br/>
<a href="{$user.take_url}">{$user.take_url}</a></p>
<p>{section name="results" loop=1 show=$user.results_priv}
You can view the results of this survey at the following URL.<br/>
<a href="{$survey.results_url}">{$survey.results_url}</a></p>
{/section}
<p>Or, you can alternatively find the survey from our Main
Page and provide your Invitation Code when prompted. The
following URL will take you to our Main Page.<br/>
<a href="{$survey.main_url}">{$survey.main_url}</a></p>
Ideally, you want to send an HTML email and an alternate, plain text version. This will allow the email to be read if the user has HTML disabled. It depends on your audience, though.
I haven't tried this, but something like this should work for both an HTML version and a plain text version.
CODE
To: {$user.email}
MIME-Version: 1.0
Subject: Invitation to Participate in Survey
Content-Type: multipart/alternative;
boundary = "--=_NextPart_000_0278_01C7A13D.F7740000"
This is a MIME encoded message.
----=_NextPart_000_0278_01C7A13D.F7740000
Content-Type: text/plain; charset=ISO-8859-1
Hello {$user.name}. You have been selected to participate in
a survey at the following site. You will need the invitation
code listed in order to access the survey.
Survey: {$survey.name}
Invitation Code: {$user.code}
The following URL already contains your Invitation Code, so
clicking on it or typing it into your browser will take
you directly to the survey.
{$user.take_url}
{section name="results" loop=1 show=$user.results_priv}
You can view the results of this survey at the following URL.
{$survey.results_url}
{/section}
Or, you can alternatively find the survey from our Main
Page and provide your Invitation Code when prompted. The
following URL will take you to our Main Page.
{$survey.main_url}
----=_NextPart_000_0278_01C7A13D.F7740000
Content-Type: text/html; charset=ISO-8859-1
<p>Hello {$user.name}. You have been selected to participate in
a survey at the following site. You will need the invitation
code listed in order to access the survey.</p>
<p>Survey: {$survey.name}
Invitation Code: {$user.code}</p>
<p>The following URL already contains your Invitation Code, so
clicking on it or typing it into your browser will take
you directly to the survey.<br/>
<a href="{$user.take_url}">{$user.take_url}</a></p>
<p>{section name="results" loop=1 show=$user.results_priv}
You can view the results of this survey at the following URL.<br/>
<a href="{$survey.results_url}">{$survey.results_url}</a></p>
{/section}
<p>Or, you can alternatively find the survey from our Main
Page and provide your Invitation Code when prompted. The
following URL will take you to our Main Page.<br/>
<a href="{$survey.main_url}">{$survey.main_url}</a></p>
<!-- HEADER SEPERATOR - DO NOT REMOVE -->
That last line still has to be there, although you're building the HTML and plain text message directly through the user of all the headers. It
should work.

---John Holmes...