There is an app in the SharePoint App Store from Microsoft for free that you add to the site. You can then add the webpart to a page and you get an interesting Contact Us page. It posts to a list which you can then set up to alert you by email.
It is interesting to see that it access a form on a different site and then posts back to your list. I am new to Office 365 so this may well be normal.
And here as an update is the step by step guide:
- Manage your public website
- Click the cog and select Site contents
- Add an app
- Select SharePoint Store on left
- Search for “contact us”
- Select the Contact Us Form and confirm it is “from Microsoft Corporation” and it is free
- Click Add It
- You will need to log in to a Microsoft account
- Accept the dialogs and trust the app
- Allow the app to download and it will be available to add to your site
- Edit the page that you want the new form on
- Click in the page where you would like the form
- Select the Insert tab
- Click to insert App Part
- In Parts select Contact Us Form
- Click Add
- Let the form load
- Edit and change the form if required
- Edit the rest of the page if required
- Save and Publish the page
- When someone enters there details it goes into a list. To access this list and get email alerts:
- On the Contact Us Form click “Contact Us Form”
- This will show you how many answers
- Click Contact Us and you will see the list of contacts (and the fields that could also be used.
- On the menu click LIST, Alert Me, Set alert on this list
- Enter the email address you want to receive the alert and click OK. You may not be able to send this to external users.
