To use InfoPath you need at least the SharePoint Online Plan 2. Not all features map to the Standard / Enterpirse on premise SKU's (Access apps are not in Standard, but are listed in SPO Plan 1). Your best place to go when trying to figure out what you have or compare against other SKU's are the service descriptions. Here is the link for SPO: https://technet.microsoft.com/en-us/library/sharepoint-online-service-description.aspx
You can still create custom forms - but it is not quite as easy as with InfoPath. Open the list in SharePoint designer. on the list setting page you see a list of all the form pages (newform.aspx, editform.aspx, dispform.asxp). You can copy these as a starting point, modify them, and set them as the default option.
MS site here has a walk thru, ti's 2010 but 2013/O365 works simliar: https://support.office.com/en-us/article/Create-a-custom-list-form-using-SharePoint-Designer-917D8FDB-EE00-4441-ADB3-A94612D1D105?ui=en-US&rs=en-US&ad=US
There are a few resources available on the web for this: https://www.bing.com/search?q=SharePoint+Designer+custom+list+form
As mentioned above the service description says you have access to Access Apps.
An additional resource may be Access Apps. You can create forms over existing lists, but you if you import your data to the access App you can use that to create forms over the data.
Here is the support link for getting started with Access apps: https://support.office.com/en-us/article/Create-an-Access-app-25F3AB3E-510D-44B0-ACCF-B976C0813E71
Based on comment
The answer is yes, it uses forms services.
With 2010 workflows you can render an InfoPath form even if the farm is only licensed for Standard features. That indicates they can be rendered without the IPFS features enabled, or having it configured as farm service.
IPFS is not a service like the traditional service applications either - you don't need to associate it with a service proxy.
I took a site and created an InfoPath form on a list. I then disabled the IPFSiteFeature explicitly and disabled enterprise features.
- As expected you can not create new InfoPath forms for lists or in form library.
- Previously created forms are still rendered thru the InfoPath browser control.
If you open SharePoint designer and look at the created pages you will see they reference the InfoPath XSN directly, that means you need something on the back end to translate that before it can be displayed in the browser, and that is what IPFS does.