We can’t see our own soul not because it doesn’t have body ( of course it doesn’t have body) but because we see only things which is not part of our being.
For Allah if we ask can we see or not it’s just the matter that we have not understood His Holy existence. I could answer your question in a single word “NO” but I know there will a question for you then how Quran has mentioned :
وُجُوهٌ يَوْمَئِذٍ نَّاضِرَةٌ
Some faces will on that Day be bright with happiness,
But here the Arabic word وُجُوهٌ doesn’t mean face, otherwise how do we get meaning of this verses:
كُلُّ مَنْ عَلَيْهَا فَانٍ
All that lives on earth or in the heavens is bound to pass away:
وَيَبْقَىٰ وَجْهُ رَبِّكَ ذُو الْجَلَالِ وَالْإِكْرَامِ
but forever will abide thy Sustainer’s Self, full of majesty and glory.
But here word وجهٌ doesn’t mean face but His Power and Might .
When Allah is everywhere, it means our own existence is not away from His Holy being therefore seeing him is not only not possible but has no sense. Can anyone define his own reality other than his body? We can't say anything about our own soul how is it long,short, thin and wide nothing while 24×7 it’s with us near us and into us but our own eyes can’t see it.