The quickest and easiest method to move your MS applications/databases online (to the cloud) is to deploy it on a hosted Windows Remote Desktop Server. We have been providing terminal server (remote desktop) hosting solutions for over 15 years. Most clients are able to deploy their Access application on a Windows Server with Remote Desktop Services within a few hours. Contact us and to discuss your options, pricing, MS licensing, etc.
As previously announced by Microsoft, Access-based web apps (Access Web Apps) and Access web databases in Office 365 and Sharepoint Online were shut down last month (April 2018). See link here: https://support.office.com/en-us/article/Access-Services-in-SharePoint-Roadmap-497fd86b-e982-43c4-8318-81e6d3e711e8?ui=en-US&rs=en-US&ad=US
You can also review our pricing calculator here: http://www.riptidehosting.com/Remote-Desktop-Hosting-Pricing.aspx
When comparing options for hosting your Access application online, ask the hosting provider if it will be a Windows Server with Remote Desktop Services or if they are hosting the Access application on a Sharepoint server. We do not provide hosting services on Sharepoint. If your MS Access application has VBA coding, it may not work in a sharepoint environment. You should also ask if your application is hosted in a dedicated OS environment (dedicated VM for you) or if it is on a shared server environment.