Improving the WHERE clause
Your EXPLAIN shows that MySQL is only utilizing one index (type_id) for selecting the rows that match the WHERE clause, even though you have multiple criteria in the clause.
To be able to utilize an index for all of the criteria in the WHERE clause, and to reduce the size of the result set as quickly as possible, add a multi-column index on the following columns on the vehicles table:
(status, date_from, date_to, type_id, price)
The columns should be in order of highest cardinality to least.
For example, vehicles.date_from is likely to have more distinct values than status, so put the date_from column before status, like this:
(date_from, date_to, price, type_id, status)
This should reduce the rows returned in the first part of the query execution, and should be demonstrated with a lower row count on the first line of the EXPLAIN result.
You will also notice that MySQL will use the multi-column index for the WHERE in the EXPLAIN result. If, by chance, it doesn't, you should hint or force the multi-column index.
Removing the unnecessary JOINs
It doesn't appear that you are using any fields in any of the joined tables, so remove the joins. This will remove all of the additional work of the query, and get you down to one, simple execution plan (one line in the EXPLAIN result).
Each JOINed table causes an additional lookup per row of the result set. So, if the WHERE clause selects 5,000 rows from vehicles, since you have 8 joins to vehicles, you will have 5,000 * 8 = 40,000 lookups. That's a lot to ask from your database server.
