I haven't been on Explorer for quite some time so can't answer your question specifically... though I'm guessing they do now, as they used to have in cabin access via plug in.
Ship's internet is generally quite expensive though, and connections are slow. If you tried to send via a web cam, you'll run up a hefty bill.
RCI does offer internet packages for the cruise. U can buy bundles of minutes for the duration of your cruise. If you are a returning crown and anchor member u should get a loyalty booklet in your room, which will have a coupon for internet use.
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Freedom of the Seas Dec 11
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Majesty of the Seas Jan 99