I solved it shortly after I posted this.

In Company.php, I changed the signature of positions() from:

    /**
     * @return \Illuminate\Database\Eloquent\Relations\HasMany
     */
    public function positions() {
        return $this->hasMany(Position::class)
                ->with(['person'=> function($query) {
                    $query->orderBy('last_name', 'asc')
                        ->orderBy('first_name', 'asc');}]
                );
    }

to:

    /**
     * @return \Illuminate\Database\Eloquent\Relations\HasMany
     */
    public function positions() {
        return $this->hasMany(Position::class)
            ->with('person')
            ->join('people', 'people.id', '=', 'positions.person_id')
            ->orderBy('people.last_name', 'asc')
            ->orderBy('people.first_name', 'asc');
    }

That correctly results in only two queries:

select * from `positions` inner join `people` on `people`.`id` = `positions`.`person_id` where `positions`.`company_id` = '124' and `positions`.`company_id` is not null order by `people`.`last_name` asc, `people`.`first_name` asc;

select * from `people` where `people`.`id` in ('1054', '1145', '1074', '1018', '1075', '1113', '1070', '1130', '1108', '1111', '1110', '1069', '1066', '1118', '1057');

I also updated positions() in Person.php so they'll be sorted by the Company name in ascending order:

    /**
     * @return \Illuminate\Database\Eloquent\Relations\HasMany
     */
    public function positions() {
        return $this->hasMany(Position::class)
            ->with('company')
            ->join('companies', 'company.id', '=', 'positions.company_id')
            ->orderBy('company.name', 'asc');
    }