How you request adult signature depends on how the carrier is connected. In most cases you turn on Signature and choose a Signature Type, either on the ship screen for one order or with shipping rules for many.
For one shipment: the ship screen
Open the order's ship screen.
Turn on Signature.
Choose a Signature Type (see the table below for which type your carrier uses).
Fetch rates and buy the label.
For many shipments: shipping rules
Create a shipping rule with conditions for the orders that need it (for example, a tag or product), and add both actions:
Set signature
Set signature type, with the adult type for your carrier
Both actions are needed. A signature type without Set signature does not request a signature. The rule action is labeled "only for shipments via Shippo/EasyPost", but USPS NSA (SKULabs) uses it too. See How to create and use shipping rules.
By carrier connection
FedEx (SKULabs / FedEx V2): go to Settings > Shipping > Carrier Accounts, edit the FedEx account, and set Signature option to Adult. Then turn on Signature on the shipment (or use the Set signature rule). FedEx uses the account's signature option and ignores the Signature Type.
USPS NSA (SKULabs): turn on Signature and set the type to Adult.
EasyPost (USPS): turn on Signature and choose Adult Signature or Adult Signature Restricted.
Shippo: turn on Signature and set the type to Adult.
UPS (SKULabs): the direct UPS connection can request signature required (turn on Signature), but not adult signature. Contact support if you need adult signature on UPS.
Adult signature usually adds a carrier fee, and the rates you see on the ship screen include it once Signature is on.