Use design tokens in GlSearchboxByType

Purpose

As part of implementing design tokens across the product, we need to update each component to use design tokens. This issue focuses on the GlSearchboxByType component from gitlab-ui.

References

• Token specification
• Vue component
• Figma component
• Total adoption scanner findings: count

Outcome

• Update the Figma Library to use the new tokens
• Update the component implementation to use the new tokens