Is there a monthly fee for using LawStream?

No. There are annual fees after the first year, but LawStream is not marketed on a “pay as you go” basis.
Annual support and upgrades are available after the first year, for an annual support fee. Beware of the apparent saving of products that charge a “modest” amount for each month. A monthly fee of $35, for example, amounts to $420 per year, and could soon exceed the cost of the one-time purchase of a LawStream package. The monthly fees are often charged “per user”, which can also quickly add up to more than you would want to pay.