We study a method to obtain the general vertex functions (of the Neveu-Schwarz sector), which are used in the covariant path integral approach to the orientable closed strings: bosonic, type-II super and heterotic strings. The regularization ambiguity of the two-point functions of string coordinates is discussed and the Weyl anomaly of them are calculated. The vertex functions are explicitly obtained for the first four mass levels and coincide with those in the operator formalism.