Yes, it is possible to do that. Instead of using camera calibration data, you can find and match features in your cameras using feature matching methods (the most famous algorithms for this purpose are SIFT and SURF). Then you can form the fundamental matrix in an optimisation process based on the matched features and epipolar geometry.  The optimisation process should minimise the error between the detected point and the epipolar line in the camera image.

Adding to Amir's answer, after you find correspondence points between you can use eight point algorithm proposed by Hartley.

https://en.wikipedia.org/wiki/Eight-point_algorithm

You can also minimized the fundamental matrix by gradient descent algorithm as in our following paper (the source code and data set are included):

https://www.researchgate.net/publication/222433232_A_neural_network_for_recovering_3D_shape_from_erroneous_and_few_depth_maps_of_shaded_images

Article A neural network for recovering 3D shape from erroneous and ...

The problem can be solved in different ways. One of the possible schemes is as follows: Projective reconstruction -> Auto-calibration -> Metric reconstruction.

If the camera intrinsics are the same for each image, then in general you need at least 3 images and at least 6 matches for auto-calibration. The minimal 6-point algorithm is described here https://www.researchgate.net/publication/249315989_A_Minimal_Six-Point_Auto-Calibration_Algorithm

Another possible way is to pre-calibrate your camera using for example Camera Calibration Toolbox from http://www.vision.caltech.edu/bouguetj/calib_doc/ 

Article A Minimal Six-Point Auto-Calibration Algorithm