Can you run a connection from the terminals on the speaker to the sub?No, absolutely not. That would result in the same issues that would occur if you connected to the amp’s speaker terminals.
However what would probably work ok, depending on the specific design of the amp, would be to connect the + inputs of the sub to the + outputs of the amp, and connect either or both of the - inputs of the sub to a chassis screw on the amp. (I’m assuming here and in what follows that one sub is being used, rather than two, and that the sub provides inputs for two channels). That is what REL recommends for their subs when connecting at speaker level to an amp having balanced or bridged outputs.
I can’t envision a design for which that approach would cause damage. The worst thing that might happen, depending on the designs of the amp and the sub, would be a loud hum. But chances are it would work ok.
An even better approach would be to solder one or both negative wires from the sub to the ground shell of an RCA plug (or optionally two plugs, using one for each negative wire), leaving the center pin unconnected, and plugging it or them into an unused RCA input connector (or connectors) on the amp. In the case of the A-S3000 and other integrated amps providing "main in" connectors, it would probably be preferable to plug in there, rather than to an input intended for a source component, since that figures to provide a more direct path to the ground of the amplifier section.
In cases where the sub provides speaker level inputs for two channels I suspect it wouldn’t make any difference whether one or both negative input terminals of the sub are connected, as the two negative input terminals of subs which provide two input channels are most likely connected directly together within the sub.
Regards,
-- Al