A group of checkboxes with the same name often hints at a many-to-many relationship - so you might want to consider that adding a vehicles table (if you don't have it already) and a user_vehicle pivot table to store the relationship.
Whenever you are expecting to be able to select multiple checkboxes, you need to send an array of selected checkboxes, therefore, the name of the input field is appended with square brackets:
<input type="checkbox" name="vehicle[]" value="Bike">Has a bike<br>
<input type="checkbox" name="vehicle[]" value="Car">Has a car<br>
<input type="checkbox" name="vehicle[]" value="Motorcycle">Has a Motorcycle
Now in your controller's store method, you will have an array for the selected options:
$request->vehicle // => array(0 => "Bike", 1 => "Motorcycle")
Lastly, if you chose the vehicles and user_vehicles tables as your preferred approach, you would be returning the relevant vehicle id's rather than the string values, then you can use the relationship's syncmethod to persist the selection (assumes you have a vehicles hasMany relationship on the User model:
<input type="checkbox" name="vehicle[]" value="1">Has a bike<br>
<input type="checkbox" name="vehicle[]" value="2">Has a car<br>
<input type="checkbox" name="vehicle[]" value="3">Has a Motorcycle
public function store(Request $request)
{
$user = App\User::create($request->only('name', 'age'); // or whatever the parent in the relationship is
$user->vehicles()->sync($request->vehicle);
}