How can I pass a JS result into a Twig template?
Per Michael's suggestion above, with some help from this answer, I passed the results from the API call back to a craft template as query params, so it looked like this (simplified): 

{% if craft.request.isAjax %}
    {% set myReps = craft.request.getParam('personId') %}
    {% set myIds = myReps | split(',') %}
    {% for myId in myIds %}
        {% set person = craft.entries.section('representatives').repId(myId).first() %}
    {% endfor %}
{% endif %}


and then output my info from there.