保険会社に連絡しますので、御名前と御住所と御電話番号を記録させていただけますか。

、。

I'll be contacting the insurance company, so could I take down your name, address, and telephone number?

Literal

Insurance-company [to-に] contact (polite) [because-ので], honor-name [and-と] honor-address [and-と] honor-phone-number [obj-を] record-causative-receive-can (polite)?

A showcase of super-polite Japanese. 御 (ご/お) is the honorific prefix elevating the listener's belongings — 御名前, 御住所, 御電話番号. ~させていただく is the hyper-humble causative-receipt frame: literally 'receive the favor of being allowed to do X,' idiomatically 'would it be OK if I...?' The potential ~ただけますか adds 'would you be able to...?' The whole ending is a mountain of polite markers for what's essentially 'can I write this down?'