No, the Foundation edition of SharePoint does not allow you to use
SharePoint Project Site Templates
or create project sites. These are available only in SharePoint 2013 Standard/Enterprise editions. If you are looking for easy task management functionality, it is available in the Foundation edition.
You have the option of upgrading to Hosted SharePoint 2013 Standard or Enterprise plan or use the task management feature of SharePoint Foundation. Hosted SharePoint Server 2013 Standard and Hosted SharePoint 2013 Enterprise are per-user based plans that enable SharePoint 2013 project site. Standard and Enterprise editions are also available in Flex Dedicated, Dedicated, and Farm modes.
The Standard or Enterprise editions of SharePoint also allow you to use SharePoint Project Site Templates and install apps from the Internal Catalog or the public SharePoint Store. Similarly, cloud-hosted and SharePoint-hosted apps can be installed on the Standard or Enterprise editions. These are not possible with the Foundation edition of SharePoint.
Best-priced SharePoint plans with Project Site Templates
If you are looking to use Project Site Templates, we offer Hosted SharePoint 2013 Standard and Hosted SharePoint Server 2013 Enterprise plans that will meet all your functionality needs and yet prove economical.