I think this hadith from sahih al-Bukhari describes a similar situattion as you mentioned:
"A woman came to the Prophet (ﷺ) offering herself to him in marriage, saying, "Have you got any interest in me (i.e. would you like to marry me?)" Anas's daughter said, "How shameless that woman was!" On that Anas said, "She is better than you, for she presented herself to Allah's Messenger (ﷺ) (for marriage).
so shyness or hiya' in such matters is out of question. Also note in marriage matters it is allowed to ask about the future spouse, and answering these questions is not considered as haram or backbiting.
The Quran is also quoting the permission for a woman to give herself to the Prophet () (33:50):
O Prophet, indeed We have made lawful to you ... and a believing woman if she gives herself to the Prophet [and] if the Prophet wishes to marry her ...
al-Qurtobi when treating this verse quoted some ahadith like this of 'Aisha and this one quoting her which show that some women did as described in the verse! Further more this hadith shows that if the women is willing to get married she can marry somebody else than the Prophet () if all agree, which can be used as general permission for woman to ask a man to marry her.
The Quran also quotes that the guardian of a lady can give or offer his daughter to such a man as you may read in the story telling us how Musa () got married in Madyan (28:26-27)!
With regard to the words of Allaah (interpretation of the meaning):
“He said: ‘I intend to wed one of these two daughters of mine to you, on condition that you serve me for eight years…’”
Here the father offered his daughter in marriage to the man. This is an established custom whereby the righteous man of Madyan offered his daughter in marriage to a righteous man of the Children of Israel, ‘Umar ibn al-Khattaab offered his daughter Hafsah in marriage to Abu Bakr and ‘Uthmaan, and the woman who offered herself in marriage offered herself to the Prophet (peace and blessings of Allaah be upon him). So it is good for a man to offer his female relative who is under his care in marriage, and for a woman to offer herself in marriage to a righteous man, following the example of the righteous salaf. Ibn ‘Umar said: When Hafsah became single (due to her husband dying), ‘Umar said to ‘Uthmaan: “If you wish, I will marry Hafsah bint ‘Umar to you.” This was narrated only by al-Bukhaari, 4005.
Tafseer al-Qurtubi, 13/271 (source islamqa #20916)
So on the whole only local customs can hinder a woman from asking the man she consider as suitable to marry her.
Also read the fatwa islamqa #99737.