Unable to access Checkout when using Storefront API

[alexMykhailov]

I am trying to make checkoutGiftCardsAppend mutation.

mutation checkoutGiftCardsAppend($checkoutId: ID!, $giftCardCodes: [String!]!) {
  checkoutGiftCardsAppend(checkoutId: $checkoutId, giftCardCodes: $giftCardCodes) {
    checkoutUserErrors {
      code
      message
    }
  }
}

When making a request I get

{
  "data": {
    "checkoutGiftCardsAppend": null
  },
  "errors": [
    {
      "message": "Unable to access Checkout.",
      "locations": [
        {
          "line": 3,
          "column": 3
        }
      ],
      "path": [
        "checkoutGiftCardsAppend"
      ]
    }
  ]
}

I do request with a proper storefront access token.

My checkout ID is of "gid://shopify/Checkout/{checkoutID}"  format.
Also, my access scopes are: 
"write_orders","write_script_tags","read_products","write_checkouts","write_themes","write_customers","write_price_rules","unauthenticated_write_checkouts","read_orders","read_script_tags","read_checkouts","read_themes","read_customers","read_price_rules","unauthenticated_read_checkouts"
Access scopes were received with /admin/oauth/access_scopes.json Get request, so they are real existing scopes.