Add active class to foundation 6 tabs while looping categories

Sure, you can use a counter. Just move your while loop a little higher. Something along these lines:

    <ul class="tabs" data-tabs id="example-tabs">
        <?php
            // counter for ul.tabs
            $i=0;
            $bulletin_types = get_object_taxonomies( 'bulletinboard' );
            foreach( $bulletin_types as $bulletin_type ) :
                    $terms = get_terms( $bulletin_type );
        foreach( $terms as $term ) :
        // increment each one
        $i++; ?>

        <li class="tabs-title
            <?php // only for the first one, add .is-active
            if($i == 1) { echo ' is-active'; } ?>
        "><a data-tabs-target="panel-<?php echo $term->slug ;?>" href="#panel-<?php echo $term->slug ;?>"><?php echo $term->name ;?></a></li>
        <?php endforeach;?>
    </ul>
    <div class="tabs-content" data-tabs-content="example-tabs">
        <?php foreach( $terms as $term ) : ?>
        <?php $bulletins = new WP_Query( array(
            'taxonomy' => $bulletin_type,
            'term' => $term->slug,
        ));?>
        <?php if( $bulletins->have_posts() ):
        // move the while up
        while( $bulletins->have_posts() ) : $bulletins->the_post();
        // reset the counter
        $i=0; ?>
        <div class="tabs-panel<?php // again add .is-active only for first
if($i==0) { echo ' is-active'; } ?>" id="panel-<?php echo $term->slug ;?>">
            <ul class="accordion" data-accordion data-allow-all-closed="true">
                <li class="accordion-item" data-accordion-item>
                    <a href="#" class="accordion-title"><?php the_title();?></a>
                    <div class="accordion-content" data-tab-content >
                        //POST CONTENT HERE
                    </div>
                </li>
                <?php endwhile; ?>
            </ul>
        </div>
        <?php endif;?>
        <?php endforeach;?>
    </div>
    <?php endforeach; ?>

Basically, you set your counter up before whatever loop you’re using – like your foreach – and then within the loop, increment it, and that way you can isolate the first item.