We use deep multi-epoch point-spread function (PSF) photometry taken with the Visible and Infrared Survey Telescope for Astronomy (VISTA) to measure and analyze the proper motions of stars within the Galactic globular cluster 47 Tucanae (47 Tuc, NGC 104). The observations are part of the ongoing near-infrared VISTA survey of the Magellanic Cloud system (VMC). The data analyzed in this study correspond to one VMC tile, which covers a total sky area of 1.77 deg(2). Absolute proper motions with respect to similar to 9070 background galaxies are calculated from a linear regression model applied to the positions of stars in 11 epochs in the K-s filter. The data extend over a total time baseline of about 17 months. We found an overall median proper motion of the stars within 47 Tuc of (mu alpha cos(delta), mu(delta)) = (+5.89 +/- 0.02 (statistical) +/- 0.13 (systematic), -2.14 +/- 0.02 (statistical) +/- 0.08 (systematic)) mas yr(-1), based on the measurements of similar to 35 000 individual sources between 5' and 42' from the cluster center. We compared our result to the proper motions from the newest US Naval Observatory CCD Astrograph Catalog (UCAC5), which includes data from the Gaia data release 1. Selecting cluster members (similar to 2700 stars), we found a median proper motion of (mu(alpha)cos(delta), mu(delta)) = (+5.30 +/- 0.03 (statistical) +/- 0.70 (systematic), -2.70 +/- 0.03 (statistical) +/- 0.70 (systematic)) mas yr(-1). Comparing the results with measurements in the literature, we found that the values derived from the VMC data are consistent with the UCAC5 result, and are close to measurements obtained using the Hubble Space Telescope. We combined our proper motion results with radial velocity measurements from the literature and reconstructed the orbit of 47 Tuc, finding that the cluster is on an orbit with a low ellipticity and is confined within the inner similar to 7.5 kpc of the Galaxy. We show that the use of an increased time baseline in combination with PSF-determined stellar centroids in crowded regions significantly improves the accuracy of the method. In future works, we will apply the methods described here to more VMC tiles to study in detail the kinematics of the Magellanic Clouds.