What vowel is used in the word เปิด?

I know some vowels have multiple parts, like เ◌ีย or เ◌ือ, but I can't find any vowel structure that includes both เ and ◌ิ. 

It's absolutely standard.  Pronounced /pə̀ət/.

 

Have a look at http://thai-notes.com/reading/lesson19.shtml for more examples, plus one common exception.

  This vowel sound is pronounced as ER as in the English word  ManagER.

The Thai vowel is written in the format that you refer to -only when there is a end consonant.

It is the ninth vowel, อา, อี, อื, อู, เอ, แอ, โอ, ออ, เออ,  
eg. To meet is เจอ, to raise up is เชิด 

It is not written เชิอด as you would expect when there is a closing consonant but เชิด where the vowel symbol ิ is used to replace the อ component.  
The short form is เออะ but it can't be shown so the tone must be learnt. เชิด long vowel, second tone (falling), in this case.