Direct answer
Usually not. A Social Security card is not an identity document, and SSA says the physical card is unnecessary for many routine needs.
When you may not need a replacement
If you know your SSN and no agency or employer specifically requires the card, replacing a lost card may not be urgent. SSA also provides secure online access to some personal record information.
When proof may still be requested
An employer, licensing body, benefits program, financial institution, or other organization may ask for documentation under its own rules. Ask which documents it accepts rather than assuming the Social Security card is the only option.
Protect the number
Do not carry the card routinely. Store it securely, avoid sending the number through insecure channels, and report suspected identity theft through the appropriate official channels.