From you description, all the interfaces are in VLAN 999, and all are configured as access ports. You have configured the VLAN interface with a /28 mask, so I presume this is a 12 port switch, as that mask only allows 14 addresses, and one of them will be the router interface for Internet access, one for the switch, and 12 for the connected devices?
In this instance, it really doesn’t matter what VLAN number you have chosen (with a very few exceptions), as none of the interfaces are trunks, this VLAN information is not carried outside of this switch, whatever the fa0/1 interface is connected to, defines which VLAN these are really in, with reference to the rest of the network. If that is a router, then this VLAN is totally isolated, and could be anything you like. You also say that all the interfaces here are in the same VLAN, so all of them can communicate with each other, and with the router, as they have Internet access.
If you cannot connect from outside of this VLAN, then you need to look at the router, and what it allows, and/or is routing for this subnet. If you cannot connect to the switch, then you need to confirm that the switch has a gateway configured to be the router IP in the same subnet (the ip default-gateway command), otherwise it has no route to get back to the subnet that you are trying to connect from.
Hope this helps. If you still have problems, then please describe the routing and other devices in the network that are connected to this switch.