Firstly, you cannot stay in the UK with an expired visa, unless you have already submitted a new visa application before the old visa expires.
Secondly, you cannot apply for a visitor visa while in the UK.
Thirdly, it would be illegal for you to intern with the company while on a visitor visa anyway, so there isn't much point in applying for one.
Unfortunately, as you cannot apply for a Tier 5 BUNAC visa from inside the UK either, it looks like you are going to have to leave the UK before your visa expires and then apply for a new visa from the US (although you would have to check if you qualify for the BUNAC visa as there are specific requirements and funds you must have for one).
The only way you would be able to get a new visa without leaving the UK is if you are getting an extension to your Tier 4 visa for a new course (and apply before your current visa expires), or if you qualify for and apply for a Tier 2 Sponsored work visa before your current visa expires. However, if your visa expires soon, it's very unlikely you would be able to sort out and apply for one of these visas before the expiry date.
Are you a US citizen? BUNAC is not available to you if you are.
Yes it is - the 6-month BUNAC Intern in Britain scheme is kind of specifically for US citizens and citizens of other countries who cannot qualify for the 2-year Tier 5 Youth Mobility Scheme.