Hi Julian,
Yes you can change the language for the web widget by going into the settings for your Zendesk Support plugin and under Web Widget Settings, add this line at the bottom of the Web Widget Code field
<script>
zE(function() {
zE.setLocale('languagecode');
});
</script>
Replace languagecode with the language code for the language you want the widget to appear in.
Also, check out https://support.zendesk.com/hc/en-us/articles/229167008-Advanced-customization-of-your-Web-Widget#topic_sty_j2r_gq for more information.