~まい?

Is the auxilary まい express the negative volitional 意向形 or is there another use? Are there any more grammar structure for negative volition?

For example: How do you express this sentence without まい?

もう橋本には何も頼むまい。

Is there anything wrong if I say like this: もう橋本には何も頼むだろうか?

The two sentences you posted have different nuances.
もう橋本には何も頼むまい -> もう橋本には何も頼まないだろう or
橋本には何も頼まないようにしよう (<--close to what you mean by negative volitional, but in a lot of contexts まい will mean 頼まないだろう)

Also, a few notes
何も is usually used with negative, so you should use 頼まない instead of 頼む (but you can say 何か頼むだろうか)
まい is usually restricted to writing, in formal contexts (newspaper editorials, etc.) so don't use it in normal conversation

Andromedashun said:

Is the auxilary まい express the negative volitional 意向形 or is there another use? Are there any more grammar structure for negative volition?

For example: How do you express this sentence without まい?

もう橋本には何も頼むまい。

Is there anything wrong if I say like this: もう橋本には何も頼むだろうか?

Click to expand...

When a verb belongs to non-volitional verbs, ～まい ALWAYS means negative presumptive ～ないだろう, whereas the ～まい form of a volitional verb can also express negative volitional ～ないようにしよう. In fact, まい is used as an indicator to check whether a verb is volitional or non-volitional.
e.g.
あの店には二度と行くまい。(行く is volitional.)
あの店には二度と行かないだろう。
or
あの店には二度と行かないようにしよう。

今日は雨は降るまい。(降る is non-volitional.)
今日は雨は降らないだろう。
☓今日は雨は降らないようにしよう。（This sentence makes no sense.）

As for your example 頼むまい, 頼む is a volitional verb, so it can mean both 頼まないだろう and 頼まないようにしよう, as tokkyo-san answered.

Thanks both for being helpful.