Hi Nirmala,
There are a number of options to do this, you can either use Odoo's builtin APIs or if you want to be more flexible you can write a an http or JSON route in Odoo to handle the request.

for eg:

# custom route for json
from odoo import http
from odoo.http import request
import json


class InvoiceBarcodeConfirm(http.Controller):

    @http.route('/invoice/barcode/confirm', type='json', auth='public', website=False)
    def confirm_invoice(self, **data):
        # assuming that the data recieved from flutter has the following structure
        # {'user_login': value,'user_pwd':password, 'invoice_barcode': barcode}
        if 'user_login' in data and 'user_pwd' in data and 'invoice_barcode' in data:
            uid = request.session.authenticate(request.env.cr.dbname, data['user_login'], data['user_pwd'])
            if uid.id:
                invoice = request.env['account.invoice'].search([('barcode', '=', data['invoice_barcode'])])
                if invoice.id:
                    invoice.action_invoice_open()
                    return json.dumps({'status': 'success'})
                else:
                    return json.dumps({'status': 'no matching invoice'})
            else:
                return json.dumps({'status': 'authentication failed'})
        else:
            return json.dumps({'status': 'not enough parameters received'})​

the code is just an illustration to how you can create a custom handler for your request.In case you just need to get your mail and attachments in odoo, you can simply configure external incoming email server from the configurations.

Hope it helped.