How do I sponsor a married sibling?

No, you just file one petition for your sibling and your sibling's spouse and children under 21 will be included in that petition. However, you should know that sibling of US citizen petitions take about 10-12 years before a visa is available so it is going to be a while.

In this case, you need to file only one Form I-130 for your sister. As a US citizen, that petition is in the Fourth family preference category, and with it, your sister, her husband and all children under 21 years can all come to the US when the petition is approved and current. Because, petitions in that category are perennially oversubscribed, note that if filed today, it will become current for at least 12 years or 2025 at the earliest. If your situation is such that your sisters' family wishes to come the US much earlier, you must consult with an experienced immigration attorney for consideration of faster immigration options for her.

No, simply file the I-130 for your sister and include her spouse and children (if any) on the same form. The processing time is about 10 years or more.

Her family can be added to the I-130.

You can file an I-130 visa petition, but it will be many years before they can apply for permanent residence. Because the 4th preference may be ending if a new law passes, you should file the petition now. You cannot file a petition for her husband, but if and when she is eligible to apply for permanent residence, her husband may be able to file with her.