You didn't provide any information about what type of array you're talking about, but the simplest answer is no, you need phase information AND element location information to be able to determine relative time of arrival and corresponding angle of arrival information.  Now, on the other hand, if you're talking about using a directional antenna and you have the ability to rotate said antenna and measure just magnitude, then you can obtain direction of arrival information from just the magnitude pattern information.   This is of course assuming line of sight (plane wave) illumination.  In all cases, multi-path AoA is more complicated.

Now, the above discussion gives you an interesting case to consider.  Assuming you're asking about the beam forming capabilities of a set of dipole elements (a typical wireless antenna system, for example) , consider the directionality of a typical linear dipole array (Yagi, Log Periodic Dipole Array, etc.).  Such antennas achieve directionality in a given frequency range based on the physical spacing of the elements and the direct summation of the received RF signal on each element along a transmission line that connects them.  The frequency dependence is a combination of the element spacing both at the propagation velocity in free space and along the transmission line.  When those phases sum properly, you obtain a single lobe along the direction of the array.  If instead I were to put a detector on each element and sum the resulting DC voltage, I would just get a higher DC voltage.  There would be no directionality, since each element would respond like a dipole to the same field regardless of direction.  (Actually there would be mutual coupling effects between the elements which DO give some directional behavior even in this case, but I'm ignoring that for now to make a point.  In fact, one directional antenna design consists of two passive resonant elements with a single dipole element in between!)  So you see, without phase information, even a common passive directional antenna wouldn't work.

I don't think you can.  From my understanding, the direction of arrival estimate will be based on the phase differences between the received signals.  Since your data is magnitude-only the phase is lost.  I only have a very low understanding of these algorithms so I don't know for sure.

You didn't provide any information about what type of array you're talking about, but the simplest answer is no, you need phase information AND element location information to be able to determine relative time of arrival and corresponding angle of arrival information.  Now, on the other hand, if you're talking about using a directional antenna and you have the ability to rotate said antenna and measure just magnitude, then you can obtain direction of arrival information from just the magnitude pattern information.   This is of course assuming line of sight (plane wave) illumination.  In all cases, multi-path AoA is more complicated.

Now, the above discussion gives you an interesting case to consider.  Assuming you're asking about the beam forming capabilities of a set of dipole elements (a typical wireless antenna system, for example) , consider the directionality of a typical linear dipole array (Yagi, Log Periodic Dipole Array, etc.).  Such antennas achieve directionality in a given frequency range based on the physical spacing of the elements and the direct summation of the received RF signal on each element along a transmission line that connects them.  The frequency dependence is a combination of the element spacing both at the propagation velocity in free space and along the transmission line.  When those phases sum properly, you obtain a single lobe along the direction of the array.  If instead I were to put a detector on each element and sum the resulting DC voltage, I would just get a higher DC voltage.  There would be no directionality, since each element would respond like a dipole to the same field regardless of direction.  (Actually there would be mutual coupling effects between the elements which DO give some directional behavior even in this case, but I'm ignoring that for now to make a point.  In fact, one directional antenna design consists of two passive resonant elements with a single dipole element in between!)  So you see, without phase information, even a common passive directional antenna wouldn't work.

As it was said before it highly depends on the antenna. For omnidirectional antennas it won't work, they are dependent on the direction only by their phase, if you take the magnitude of the array response you get a constant over angle and hence any directional information is lost.

On the other hand, if you consider, say, a circular array of directional antennas you can estimate the DoA. The estimate may not be very accurate but this depends on the number of antennas and their directivity. The simplest estimate would be to choose the angle corresponding to the antenna that sees the highest magnitude. A better approach would be to consider all the antennas that have non-negligible magnitudes and interpolate between them.

Thank you Mr. Matthew, Mr.Michael and Mr. Florian for your insights!. 

In the literature of FFT beamforming, the criteria to be considered is the main lobe HPBW should be exactly at the nulls of the sidelobes of the adjacent FFT bins. Main lobes width overlaps(for high HPBW)  when two closely spaced angles are considered (i.e here adjacent FFT bins)

The issue is that the FFT still requires phase information.  It's just a mathematical way of summing independent signals vs. doing so electrically.  Expanding on Florian's post a bit, his array of directional antennas is just a way to electrically steer a directional beam vs. mechanically steering it with a positioner.  In either case, you steer, measure, steer, measure, etc.  Note that you could also accomplish the task with an array of dipoles where the signals are all combined through phase shifters, creating an electrically steerable array.  You don't need to worry about the phase information at the receiver because you're steering in the RF/analog domain.  The advantage is you need only one receiver, but again, you have a time dependent process of steering the beam and sampling the result.  The advantage of the FFT (or any digital beam steering approach) is that you only need to sample once and you can run through the math repeatedly with any desired phase relationship to obtain the directional information.  However, this comes at the cost of multiple receivers.  On the transmit side you don't really gain anything over the phase shifter approach, assuming of course that you have independent control of the phase and magnitude at each antenna.

Yes, FFT beamforming is surely solve your issue. By using FFT we can clearly estimate the spectrum.

Go head all the best