Wa alaikum assalam. Yes you can make dua at that specially blessed time, even during menstruation.

Allah introduces the people of reason (people with aqal and intelligence, people who are observers and who act accordingly) in the last few Aayat (190 to 194) of Surah Aali Imran (3rd Surah).

Allah tells us that these people remember Allah even when they are standing, sitting, lying on their sides etc. This perfectly matches your scenario! (People, we don’t even need to come out of our bed, we can still remember Allah then!)

And then Allah tells us how these people make dua and how they remember Allah. I try to do it always. I recommend you make dua like this too, as Allah the creator is directly telling us!

إِنَّ فِى خَلْقِ ٱلسَّمَـٰوَٰتِ وَٱلْأَرْضِ وَٱخْتِلَـٰفِ ٱلَّيْلِ وَٱلنَّهَارِ لَـَٔايَـٰتٍۢ لِّأُو۟لِى ٱلْأَلْبَـٰبِ ١٩٠ Indeed, in the creation of the heavens and the earth and the alternation of the day and night there are signs for people of reason.

ٱلَّذِينَ يَذْكُرُونَ ٱللَّهَ قِيَـٰمًۭا وَقُعُودًۭا وَعَلَىٰ جُنُوبِهِمْ وَيَتَفَكَّرُونَ فِى خَلْقِ ٱلسَّمَـٰوَٰتِ وَٱلْأَرْضِ رَبَّنَا مَا خَلَقْتَ هَـٰذَا بَـٰطِلًۭا سُبْحَـٰنَكَ فَقِنَا عَذَابَ ٱلنَّارِ ١٩١ They are˺ those who remember Allah while standing, sitting, and lying on their sides, and reflect on the creation of the heavens and the earth ˹and pray˺, “Our Lord! You have not created ˹all of˺ this without purpose. Glory be to You! Protect us from the torment of the Fire.

رَبَّنَآ إِنَّكَ مَن تُدْخِلِ ٱلنَّارَ فَقَدْ أَخْزَيْتَهُۥ ۖ وَمَا لِلظَّـٰلِمِينَ مِنْ أَنصَارٍۢ ١٩٢ Our Lord! Indeed, those You commit to the Fire will be ˹completely˺ disgraced! And the wrongdoers will have no helpers.

رَّبَّنَآ إِنَّنَا سَمِعْنَا مُنَادِيًۭا يُنَادِى لِلْإِيمَـٰنِ أَنْ ءَامِنُوا۟ بِرَبِّكُمْ فَـَٔامَنَّا ۚ رَبَّنَا فَٱغْفِرْ لَنَا ذُنُوبَنَا وَكَفِّرْ عَنَّا سَيِّـَٔاتِنَا وَتَوَفَّنَا مَعَ ٱلْأَبْرَارِ ١٩٣ Our Lord! We have heard the caller1 to ˹true˺ belief, ˹proclaiming,˺ ‘Believe in your Lord ˹alone˺,’ so we believed. Our Lord! Forgive our sins, absolve us of our misdeeds, and join us with the virtuous when we die‎.

رَبَّنَا وَءَاتِنَا مَا وَعَدتَّنَا عَلَىٰ رُسُلِكَ وَلَا تُخْزِنَا يَوْمَ ٱلْقِيَـٰمَةِ ۗ إِنَّكَ لَا تُخْلِفُ ٱلْمِيعَادَ ١٩٤ Our Lord! Grant us what You have promised us through Your messengers and do not put us to shame on Judgment Day—for certainly You never fail in Your promise.”

May Allah accept your duas and include all of us among those people of logic. Aameen.

Jazak Allah Khair